Diese Formelsammlung fasst Formeln und Definitionen der Tensoralgebra für Tensoren zweiter Stufe in der Kontinuumsmechanik zusammen. Es wird der dreidimensionale Raum zugrunde gelegt.
- Operatoren wie
werden nicht kursiv geschrieben.
- Buchstaben die als Indizes benutzt werden:
.
Ausnahme:
Die imaginäre Einheit
und die #Vektorinvariante
werden in Abgrenzung zu den Indizes nicht kursiv geschrieben.
![{\displaystyle p,q,r,s\in \{1,2,\ldots ,9\}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82z2oQzgvFaga3atmQnja3yghBoqdCaqzEnteNyjdFnDlBoAzBzqaO)
![{\displaystyle u,v\in \{1,2,\ldots ,6\}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83yqi3o2rAzts0zNdEyqdAyjBBnqeQoDmPoNiQaNmOzjnFyjo4njzA)
- Alle anderen Buchstaben stehen für reelle Zahlen oder komplexe Zahlen.
- Vektoren:
- Alle hier verwendeten Vektoren sind geometrische Vektoren im dreidimensionalen euklidischen Vektorraum
.
- Vektoren werden mit Kleinbuchstaben bezeichnet.
Ausnahme #Dualer axialer Vektor ![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PajiNntG2ytsPyja1ytC2aNaQyta3yjw1zNBDzDCPaDCNzNBDz2a2)
- Einheitsvektoren mit Länge eins werden wie in ê mit einem Hut versehen. Die Standardbasis von
ist ê1,2,3.
- Vektoren mit unbestimmter Länge werden wie in
mit einem Pfeil versehen.
- Dreiergruppen von Vektoren wie in
oder
bezeichnen eine rechtshändige Basis von
.
- Gleichnamige Basisvektoren mit unterem und oberem Index sind dual zueinander, z. B.
ist dual zu
.
- Tensoren zweiter Stufe werden wie in A mit fetten Großbuchstaben notiert. Die Menge aller Tensoren wird mit
bezeichnet. Tensoren höherer Stufe werden mit einer hochgestellten Zahl wie in
geschrieben. Tensoren vierter Stufe sind Elemente der Menge
.
- Es gilt die Einstein'sche Summenkonvention ohne Beachtung der Indexstellung.
- Kommt in einer Formel in einem Produkt ein Index doppelt vor wie in
wird über diesen Index summiert:
.
- Kommen mehrere Indizes doppelt vor wie in
wird über diese summiert:
.
- Ein Index, der nur einfach vorkommt wie
in
, ist ein freier Index. Die Formel gilt dann für alle Werte der freien Indizes:
.
Formelzeichen |
Abschnitt in der Formelsammlung |
Wikipedia-Artikel
|
![{\displaystyle \mathrm {Sp,tr,I} _{1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ajmPaNnAatdCygwOaNKNaNK2yjBAaNzCaqrCzAo1oNnBaDJAajFD) |
#Spur
|
Spur (Mathematik), Hauptinvariante
|
![{\displaystyle \mathrm {I} _{2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zDs1z2dFotdCa2hFyjaPyqs1ytm1aqnFa2i1o2sQzjlEzNlCaNm0) |
#Zweite Hauptinvariante
|
Hauptinvariante
|
|
#Determinante
|
Determinante, Hauptinvariante
|
sym |
#Symmetrischer Anteil |
Symmetrische Matrix
|
skw, skew |
#Schiefsymmetrischer Anteil |
Schiefsymmetrische Matrix
|
adj |
#Adjunkte |
Adjunkte
|
cof |
#Kofaktor |
Minor (Mathematik)#Kofaktormatrix
|
dev |
#Deviator |
Deviator, Spannungsdeviator
|
sph |
#Kugelanteil |
Kugeltensor
|
Formelzeichen |
Elemente
|
![{\displaystyle \mathbb {R} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83yjo4zjBDzNo1ngi3otG0ngdDa2zDntwNntC2oqhCztFFztG2ygvD) |
Reelle Zahlen
|
![{\displaystyle \mathbb {C} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AyqhCnjwQyjsQytrEyqe2nje4njm0zqnCyqa5aAaOoNm5nDs0yqi3) |
Komplexe Zahlen
|
![{\displaystyle \mathbb {V} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AnDvBnDw3zDBBaqaNaqe4zDK5aDi0ago0ygw5yqdFyjzBoAwOaNs4) |
Vektoren
|
|
Tensoren zweiter Stufe
|
|
#Tensoren vierter Stufe
|
![{\displaystyle \delta _{ij}=\delta ^{ij}=\delta _{i}^{j}=\delta _{j}^{i}={\begin{cases}1&\mathrm {falls} \quad i=j\\0&\mathrm {sonst} \end{cases}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81yjw4oDm4aji4njePaNC5ngzFzjzAzDoPngrCztCNyjhAzAa2ags1)
Für Summen gilt dann z. B.
![{\displaystyle v_{i}\delta _{ij}=v_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AaAwQaAdCnAo2zNaPztrDztwPaAe2zti5oqrCygdDztmPoDw4oDFC)
![{\displaystyle A_{ij}\delta _{ij}=A_{ii}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DnjiQatmOzDnCaDCPygnAnta0zNFCzAiOoNmOyjrBzghAatC2nAa4)
Dies gilt für die anderen Indexgruppen entsprechend.
![{\displaystyle \epsilon _{ijk}={\hat {e}}_{i}\cdot ({\hat {e}}_{j}\times {\hat {e}}_{k})={\begin{cases}1&{\text{falls}}\;(i,j,k)\in \{(1,2,3),(2,3,1),(3,1,2)\}\\-1&{\text{falls}}\;(i,j,k)\in \{(1,3,2),(2,1,3),(3,2,1)\}\\0&{\text{sonst, d.h. bei doppeltem Index}}\end{cases}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zjhBzDo2yto1aNw0z2sOagw3zDdEnDeOzNhDaNGOz2sOaNFFytrD)
![{\displaystyle \epsilon _{ijk}\epsilon _{lmn}={\begin{vmatrix}\delta _{il}&\delta _{jl}&\delta _{kl}\\\delta _{im}&\delta _{jm}&\delta _{km}\\\delta _{in}&\delta _{jn}&\delta _{kn}\end{vmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80oDm2zjK0agaPzAzEaNFAoArDytG2zNGOzDdEnDzBztG3oDBAztJE)
![{\displaystyle \epsilon _{ijk}\epsilon _{klm}=\delta _{il}\delta _{jm}-\delta _{im}\delta _{jl}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QzDG4njBDaAo3yqs2aDKPztBDztCNzDBAoqi0athAyti4ytmQnDJA)
![{\displaystyle \epsilon _{ijk}\epsilon _{jkl}=2\delta _{il}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DatBAngw0ytnFoqaPoDhEoDw5ajdCyjCNzqdDots3ajm2njK2yje3)
![{\displaystyle \epsilon _{ijk}\epsilon _{ijk}=6}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85zDK0oDvDaNrDnqrAoAdFzgw5nDGQzNBAaqoQzDaOzqa3z2vCntK4)
Kreuzprodukt:
![{\displaystyle a_{i}{\hat {e}}_{i}\times b_{j}{\hat {e}}_{j}=\epsilon _{ijk}a_{i}b_{j}{\hat {e}}_{k}=\epsilon _{ijk}a_{j}b_{k}{\hat {e}}_{i}=\epsilon _{ijk}a_{k}b_{i}{\hat {e}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FagsQzgzDoNFFago0zqzFzDKPnqzEnqe4zqeNyqaQzjBDajCOaDK5)
![{\displaystyle \epsilon _{ijk}{\hat {e}}_{k}={\hat {e}}_{i}\times {\hat {e}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84oqeNzAa1nDhFygi2ztlEo2nAzjoPoNhDagaOaDJBnAi4oDhCntKP)
Die hier verwendeten Vektoren sind Spaltenvektoren
![{\displaystyle {\vec {a}}=a_{i}{\hat {e}}_{i}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnAhCatzEatvBaAvDaqvAzgi1oDaPztoPnqaOzNo2agw4zje1nDC4)
Drei Vektoren
können spaltenweise in einer 3×3-Matrix
arrangiert werden:
![{\displaystyle M={\begin{pmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\end{pmatrix}}={\begin{pmatrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NotzCotrFagsQyqo3yta2oqi2nAa1nAo5zDsOoNm3aDe2atBCnjm0)
Die Determinante der Matrix
![{\displaystyle |M|={\begin{vmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\end{vmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81oAi3aAw1njlEothEajm0z2aNztnAnjhFaDaQygvFyqdFyghBzDa4)
ist
Also gewährleistet
, dass die Vektoren
eine rechtshändige Basis bilden.
Die Spaltenvektoren bilden eine Orthonormalbasis, wenn
![{\displaystyle M^{\top }M={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81nDdAaDw2zgvDaNw5oDhEyqrEoNBEnAnCoDi3oqvEaqe4njlEoAzF)
worin
die transponierte Matrix ist. Bei der hier vorausgesetzten Rechtshändigkeit gilt dann zusätzlich
.
Basisvektoren
Duale Basisvektoren
Beziehungen zwischen den Basisvektoren
![{\displaystyle {\vec {g}}_{i}\cdot {\vec {g}}^{j}=\delta _{i}^{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zNmQyqrEzNG3zNC4oAs5ajaQnqaQaDnCzthBnDm4njBCzqi3aqe4)
![{\displaystyle {\vec {g}}^{1}={\frac {{\vec {g}}_{2}\times {\vec {g}}_{3}}{({\vec {g}}_{1},{\vec {g}}_{2},{\vec {g}}_{3})}},\quad g^{2}={\frac {{\vec {g}}_{3}\times {\vec {g}}_{1}}{({\vec {g}}_{1},{\vec {g}}_{2},{\vec {g}}_{3})}},\quad g^{3}={\frac {{\vec {g}}_{1}\times {\vec {g}}_{2}}{({\vec {g}}_{1},{\vec {g}}_{2},{\vec {g}}_{3})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zqnAnDrFnDe2nDmQztzCoti1otvBate5aDa3aga0z2zBnjs4atJB)
![{\displaystyle {\vec {g}}_{1}={\frac {{\vec {g}}^{2}\times {\vec {g}}^{3}}{({\vec {g}}^{1},{\vec {g}}^{2},{\vec {g}}^{3})}},\quad g_{2}={\frac {{\vec {g}}^{3}\times {\vec {g}}^{1}}{({\vec {g}}^{1},{\vec {g}}^{2},{\vec {g}}^{3})}},\quad g_{3}={\frac {{\vec {g}}^{1}\times {\vec {g}}^{2}}{({\vec {g}}^{1},{\vec {g}}^{2},{\vec {g}}^{3})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EajrCntw2oDo5aAw3zDs2ztlBo2aQo2eOaDG1a2rFotdDnAw0nAwO)
mit dem Spatprodukt
![{\displaystyle ({\vec {a}},{\vec {b}},{\vec {c}}):={\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})={\vec {c}}\cdot ({\vec {a}}\times {\vec {b}})={\vec {b}}\cdot ({\vec {c}}\times {\vec {a}})={\begin{vmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\end{vmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FaqeQoNvBota0yjo2oqoNnAhCyga1age1ygdEyjoPnge1yqw5ajK2)
Trägt man die Basisvektoren spaltenweise in eine Matrix ein, dann finden sich die dualen Basisvektoren in den Zeilen der Inversen oder den Spalten der #transponiert #Inversen
:
![{\displaystyle {\begin{pmatrix}{\vec {g}}^{1}&{\vec {g}}^{2}&{\vec {g}}^{3}\end{pmatrix}}={\begin{pmatrix}{\vec {g}}_{1}&{\vec {g}}_{2}&{\vec {g}}_{3}\end{pmatrix}}^{\top -1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82oqo4ygsPajm0ytwOo2hCztiQnghEzjoQaNe5aqzAzja2aNm5aqsO)
In der Standardbasis wie in jeder Orthonormalbasis sind die Basisvektoren
zu sich selbst dual:
![{\displaystyle {\hat {e}}_{i}={\hat {e}}^{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DztaQoAa1a2o4ajlCztG5zNa4nte1yqdFajw3agi2a2vEoDe5yqs0)
![{\displaystyle {\vec {v}}=v_{i}{\hat {e}}_{i}\quad \rightarrow \;v_{i}={\vec {v}}\cdot {\hat {e}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnjG1yqi1oAa0zNlEzDo4njKQzAiQaDnFzDJAoqzAotdBzDGNzgo3)
![{\displaystyle {\vec {v}}=v^{i}{\vec {g}}_{i}\quad \rightarrow \;v^{i}={\vec {v}}\cdot {\vec {g}}^{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Czts3zAzBnjhEaqe4ngnFzAnCyga1ztnFnja0oDG0aqeNzDGOajs5)
![{\displaystyle {\vec {v}}=v_{i}{\vec {g}}^{i}\quad \rightarrow \;v_{i}={\vec {v}}\cdot {\vec {g}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DoNw2yqdDntG2zjvFzNrAnAe3ytwNo2s3aDw3aNzAyqaNz2wNatsQ)
![{\displaystyle ({\vec {g}}_{i}\cdot {\vec {g}}_{k})({\vec {g}}^{k}\cdot {\vec {g}}^{j})={\vec {g}}_{i}\cdot ({\vec {g}}^{j}\cdot {\vec {g}}^{k}){\vec {g}}_{k}={\vec {g}}_{i}\cdot {\vec {g}}^{j}=\delta _{i}^{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80nte4nDe4ztC1aqoOoDwPajzCatvDz2aPnAhEaArAoDzEatCPyjJF)
Wechsel von
Basis
mit dualer Basis
nach
Basis
mit dualer Basis
:
![{\displaystyle {\vec {v}}=v_{i}\,{\vec {g}}^{i}=v_{i}^{\ast }\,{\vec {h}}^{i}\quad \rightarrow \;v_{i}^{\ast }=({\vec {h}}_{i}\cdot {\vec {g}}^{j})v_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DaNBBatG1aArBztwNotG4yqsOyqa4zDe4aqdCatJCztrDzAw0nqoQ)
Matrizengleichung:
![{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{1}^{\ast }\\v_{2}^{\ast }\\v_{3}^{\ast }\end{pmatrix}}=&{\begin{pmatrix}{\vec {h}}_{1}\cdot {\vec {g}}^{1}&{\vec {h}}_{1}\cdot {\vec {g}}^{2}&{\vec {h}}_{1}\cdot {\vec {g}}^{3}\\{\vec {h}}_{2}\cdot {\vec {g}}^{1}&{\vec {h}}_{2}\cdot {\vec {g}}^{2}&{\vec {h}}_{2}\cdot {\vec {g}}^{3}\\{\vec {h}}_{3}\cdot {\vec {g}}^{1}&{\vec {h}}_{3}\cdot {\vec {g}}^{2}&{\vec {h}}_{3}\cdot {\vec {g}}^{3}\end{pmatrix}}{\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}\\=&{\begin{pmatrix}{\vec {h}}_{1}&{\vec {h}}_{2}&{\vec {h}}_{3}\end{pmatrix}}^{\top }{\begin{pmatrix}{\vec {g}}^{1}&{\vec {g}}^{2}&{\vec {g}}^{3}\end{pmatrix}}{\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Na2sNytFFztw1zDKPoDe2aNrFntoQzqiQoAvAzNJDyjs5yjKPzge3)
Die grundlegenden Eigenschaften des dyadischen Produkts „⊗“ sind:
Abbildung
![{\displaystyle {\vec {a}}\otimes {\vec {g}}=\mathbf {T} \in {\mathcal {L}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AztwOz2a0aNzCzNzCnDzEaje1njsNyjGOztlCyge0a2w5ntKPajK0)
Multiplikation mit einem Skalar:
![{\displaystyle x({\vec {a}}\otimes {\vec {g}})=(x{\vec {a}})\otimes {\vec {g}}={\vec {a}}\otimes (x{\vec {g}})=x{\vec {a}}\otimes {\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzDsQyqw2yji2nAo2zDs2aDnEoqhBzjeNa2zBatCNoDrEotG3aAaP)
Distributivität:
![{\displaystyle (x+y){\vec {a}}\otimes {\vec {g}}=x{\vec {a}}\otimes {\vec {g}}+y{\vec {a}}\otimes {\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zAw5oqa1nqsQate1ajm5ntKNoAaOzDBAo2i3ygs4atiQnDoOzDe5)
![{\displaystyle ({\vec {a}}+{\vec {b}})\otimes {\vec {g}}={\vec {a}}\otimes {\vec {g}}+{\vec {b}}\otimes {\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ajC3oNs4yjrAzDBAnAaPyjG2ate3zjo0oNa0nji2zgdCzNe2yjnE)
![{\displaystyle {\vec {a}}\otimes ({\vec {g}}+{\vec {h}})={\vec {a}}\otimes {\vec {g}}+{\vec {a}}\otimes {\vec {h}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Aaqw5otrAnti1njo5ygeOzAi3oDw3atm2zgeOaAzBytnCaNa2yjwP)
Skalarprodukt:
![{\displaystyle ({\vec {a}}\otimes {\vec {g}}):({\vec {b}}\otimes {\vec {h}})=({\vec {a}}\cdot {\vec {b}})({\vec {g}}\cdot {\vec {h}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CyjnBygdData5nji5aAdEntrEntiPagdDajoPzAiNoqzBzts2otsQ)
Weitere Eigenschaften von Dyaden siehe #Dyade und den folgenden Abschnitt.
Durch die Eigenschaften des dyadischen Produktes wird
zu einem euklidischen Vektorraum und entsprechend kann jeder Tensor komponentenweise bezüglich einer Basis von
dargestellt werden:
mit Komponenten
.
Die Dyaden
und
bilden Basissysteme von
.
Abbildung
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})^{\top }:={\vec {g}}\otimes {\vec {a}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EoAs1atCPnDG3ngw2oNJAoNG4oti5o2aQaNi4agiQotBAzAdBaNa1)
![{\displaystyle (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})^{\top }=A_{ij}({\hat {e}}_{j}\otimes {\hat {e}}_{i})=A_{ji}({\hat {e}}_{i}\otimes {\hat {e}}_{j})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaAs3nAa1nDBEnjJCaNGPaNm2oqvEoAiQoNmNztm0zqzAatBAa2i4)
![{\displaystyle (A^{ij}{\vec {a}}_{i}\otimes {\vec {g}}_{j})^{\top }=A^{ij}({\vec {g}}_{j}\otimes {\vec {a}}_{i})=A^{ji}({\vec {g}}_{i}\otimes {\vec {a}}_{j})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzNzDnqhDnqaQz2eNoNe3a2nBaNo0z2s1zDsNzqdEyjaOoDa0oDm5)
![{\displaystyle \left(\mathbf {A} ^{\top }\right)^{\top }=\mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81zDmOaqe4ztw5yte4yjnCaDzEzNdFnqnEoNBDyjw2yjs5nDhFzjnB)
![{\displaystyle (\mathbf {A+B} )^{\top }=\mathbf {A} ^{\top }+\mathbf {B} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zjdEzAe5yjKOoNm5aDe5ajs2atzCzNGNzga5yja5oNdDo2i3ytw4)
![{\displaystyle (\mathbf {A\cdot B} )^{\top }=\mathbf {B} ^{\top }\cdot \mathbf {A} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BajzEzji5z2i3zjw0oqsPoDlFati5ntG1ngaPzDhBaDC5atFAzjvB)
Abbildung
oder
Dyaden:
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot {\vec {h}}:=({\vec {g}}\cdot {\vec {h}}){\vec {a}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaqhDzti2zjBCyjFFyjhAnDhAajrBnqhBnjw0z2a3nAo4otBAoDnF)
![{\displaystyle {\vec {b}}\cdot ({\vec {a}}\otimes {\vec {g}}):=({\vec {a}}\cdot {\vec {b}}){\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85zge5oDnFytJAajvBygvCajo0atGNzgnDoDKNnqwOoDa3zDoQyqeP)
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot {\vec {h}}={\vec {h}}\cdot ({\vec {a}}\otimes {\vec {g}})^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zgrBzgnEnDCQygrBajBFnDi2zqsQnDw2zjiPnAs5nqs5oDe0oDK5)
![{\displaystyle {\vec {b}}\cdot ({\vec {a}}\otimes {\vec {g}})=({\vec {a}}\otimes {\vec {g}})^{\top }\cdot {\vec {b}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80otzCagsPaNnCzjoNoAw0zDBFyqrEoAvAaDo5zNhBaNi2ajBEatlC)
Allgemeine Tensoren:
![{\displaystyle A_{ij}({\hat {e}}_{i}\otimes {\hat {e}}_{j})\cdot {\vec {v}}=A_{ij}({\vec {v}}\cdot {\hat {e}}_{j}){\hat {e}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DzqvEotaOnAiNzNaOngrDoAsNajlDoNzEots4ygvBajwPzDm4aNvA)
![{\displaystyle A^{ij}({\vec {a}}_{i}\otimes {\vec {g}}_{j})\cdot {\vec {v}}=A^{ij}({\vec {v}}\cdot {\vec {g}}_{j}){\vec {a}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81zDa5ntw3o2o2aDBAaqdCygdEnjFEytmPoqi0nqw5zDoPaDJCythF)
![{\displaystyle {\vec {v}}\cdot A_{ij}({\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{ij}({\vec {v}}\cdot {\hat {e}}_{i}){\hat {e}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84aghAzNiOyjhBatCQzNsOyqwPzNm0otnEz2vFoqvAngw4otFEoNm3)
![{\displaystyle {\vec {v}}\cdot A^{ij}({\vec {a}}_{i}\otimes {\vec {g}}_{j})=A^{ij}({\vec {v}}\cdot {\hat {a}}_{i}){\vec {g}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaNs1zjzBzja5yqwQa2nCoqzEoNzEajw5oNCOo2s4yqhBytFCnja0)
Symbolisch:
![{\displaystyle \mathbf {A} \cdot {\vec {v}}={\vec {v}}\cdot \mathbf {A} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pzgw2ajJCoNiNyqwQyjCQotBFoNiQnqhFnAoOzNJCnjeQagePzAs1)
![{\displaystyle {\vec {v}}\cdot \mathbf {A} =\mathbf {A} ^{\top }\cdot {\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84aNmOoDaQote1oDi4yjdEztrDaNoOygaPoAsPoDw0zNs1agvEnAsO)
Abbildung
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot ({\vec {h}}\otimes {\vec {u}}):=({\vec {g}}\cdot {\vec {h}}){\vec {a}}\otimes {\vec {u}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85atJFzge2nAi2aDa1agwOotBFoNGNatvAzDaNyta3zAe4zqwPaNCP)
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot \mathbf {A} ={\vec {a}}\otimes ({\vec {g}}\cdot \mathbf {A} )={\vec {a}}\otimes {\vec {g}}\cdot \mathbf {A} ={\vec {a}}\otimes (\mathbf {A} ^{\top }\cdot {\vec {g}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzAo1zgiQaAnEo2wQyqvDo2i4a2eQzqe2o2rCajK2ztmNnqoPnqhB)
![{\displaystyle \mathbf {A} \cdot ({\vec {a}}\otimes {\vec {g}})=(\mathbf {A} \cdot {\vec {a}})\otimes {\vec {g}}=\mathbf {A} \cdot {\vec {a}}\otimes {\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzDvCzgs4aDCQaAdBajJCytKNnjm2ngzDoDdEntvDatFAoqsQyjw5)
![{\displaystyle (A_{ik}{\hat {e}}_{i}\otimes {\hat {e}}_{k})\cdot (B_{lj}{\hat {e}}_{l}\otimes {\hat {e}}_{j})=A_{ik}B_{kj}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81oNmQzthCntvAzNKQnAdFztzAzAo1ntw4njCOnDK2ajhBaNs4nDo2)
![{\displaystyle \left(A^{ij}{\vec {a}}_{i}\otimes {\vec {g}}_{j}\right)\cdot \left(B^{kl}{\vec {h}}_{k}\otimes {\vec {u}}_{l}\right)=A^{ij}({\vec {g}}_{j}\cdot {\vec {h}}_{k})B^{kl}{\vec {a}}_{i}\otimes {\vec {u}}_{l}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zAs2zjlFa2s4aDwQote4aDa4zAe4ztiNzDmNatdEaNhEa2oPzjK4)
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Definition über die #Spur:
![{\displaystyle ({\vec {a}}\otimes {\vec {g}}):({\vec {b}}\otimes {\vec {h}}):=\mathrm {Sp} (({\vec {a}}\otimes {\vec {g}})^{\top }\cdot ({\vec {b}}\otimes {\vec {h}}))=({\vec {a}}\cdot {\vec {b}})({\vec {g}}\cdot {\vec {h}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzDrCoDo5a2rEotrFzAe5aqe3nAwPngo5oDw1yqo2zDnDoqzDntrD)
![{\displaystyle \mathbf {A} :\mathbf {B} :=\mathrm {Sp} (\mathbf {A} ^{\top }\cdot \mathbf {B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AotG2ntm1agsPaDBDzgeOzjhDytiNaAdEnAs1zNFFoDnEzAzEnqo4)
Eigenschaften:
![{\displaystyle \mathbf {A} :\mathbf {B} =\mathbf {B} :\mathbf {A} =\mathbf {A} ^{\top }:\mathbf {B} ^{\top }=\mathbf {B} ^{\top }:\mathbf {A} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzgnEaNeOatC4z2zDngiNzNaPzge3nDeNajGQotrCnDoOnjFEnja1)
![{\displaystyle \mathbf {A} ^{\top }:\mathbf {B} =\mathbf {A} :\mathbf {B} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzgdDnqsOzDvAnjBDyqdBoNwPo2nCzDa4aAa4nqzEyjs0zqzBnDhE)
![{\displaystyle \mathbf {A} :(\mathbf {B\cdot C} )=(\mathbf {B} ^{\top }\cdot \mathbf {A} ):\mathbf {C} =(\mathbf {A\cdot C} ^{\top }):\mathbf {B} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85zAvBaghDzDG1oAzCzNrAnqnDoNwPoNnBatvDaghAoDdCntrBaNzF)
![{\displaystyle (\mathbf {A\cdot B} ):\mathbf {C} =\mathbf {B} :(\mathbf {A} ^{\top }\cdot \mathbf {C} )=\mathbf {A} :(\mathbf {C\cdot B} ^{\top })}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PygzFajs5z2rEnDhAa2dCz2vEaDs4oNnFotvEotdEnAvCoDdAzjsN)
![{\displaystyle ({\vec {u}}\otimes {\vec {v}}):\mathbf {A} ={\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zNw1zjwOotrAnqhBztwQajwQnDa3aNi2ytrBagiQaNzAytFDzgo1)
Abbildung
oder
Dyaden:
![{\displaystyle {\vec {a}}\times ({\vec {b}}\otimes {\vec {g}})=({\vec {a}}\times {\vec {b}})\otimes {\vec {g}}={\vec {a}}\times {\vec {b}}\otimes {\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaNdCntK1ztsQoNGPaDC1aDC0nji1oNs4zqrDngw5nDJBzjw0ntK0)
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\times {\vec {h}}={\vec {a}}\otimes ({\vec {g}}\times {\vec {h}})={\vec {a}}\otimes {\vec {g}}\times {\vec {h}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85otG0zqzEaqsNajaQngdEoqrAaNa4atFAaNeQotBEoDa1njzDytC4)
![{\displaystyle {\vec {a}}\times {\vec {b}}\otimes {\vec {g}}=-[({\vec {b}}\otimes {\vec {g}})^{\top }\times {\vec {a}}]^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84nqsQzjs0zjJEzDnBzDoPnDK4ntw4zqnEzthAatoNotw3yjKPyqo3)
![{\displaystyle {\vec {a}}\otimes {\vec {g}}\times {\vec {h}}=-[{\vec {h}}\times ({\vec {a}}\otimes {\vec {g}})^{\top }]^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OajK4oNvBnDJAagw1zjaPzgzAnteQoAzFntG1aDBDnAw5ztKQzqrA)
![{\displaystyle a_{j}{\hat {e}}_{j}\times (A_{kl}{\hat {e}}_{k}\otimes {\hat {e}}_{l})=a_{j}A_{kl}({\hat {e}}_{j}\times {\hat {e}}_{k})\otimes {\hat {e}}_{l}=\epsilon _{ijk}a_{j}A_{kl}{\hat {e}}_{i}\otimes {\hat {e}}_{l}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82yjGPzNw5ntzFnDFEztCOatJBoNmQaNG0zDi1oqnBzAa5nAo2aNzB)
![{\displaystyle (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})\times a_{k}{\hat {e}}_{k}=A_{ij}a_{k}{\hat {e}}_{i}\otimes ({\hat {e}}_{j}\times {\hat {e}}_{k})=\epsilon _{jkl}A_{ij}a_{k}{\hat {e}}_{i}\otimes {\hat {e}}_{l}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PoAoNyto1ngoQaji0zAiPoArAygrCoqs5a2a1zqnDoDBEo2aPnto0)
Allgemeine Tensoren:
![{\displaystyle ({\vec {a}}\times \mathbf {A} )\cdot {\vec {g}}:={\vec {a}}\times (\mathbf {A} \cdot {\vec {g}})={\vec {a}}\times ({\vec {g}}\cdot \mathbf {A} ^{\top })}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QotwOoDdEaNmNoNC4ajlDoti4nDdBzDm0agdBnDdAoAo5yje1ztnA)
![{\displaystyle {\vec {b}}\cdot ({\vec {a}}\times \mathbf {A} ):=({\vec {b}}\times {\vec {a}})\cdot \mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnAs3oDBFaNBBzNFDnqrCaqi3zAi1zjw5agnCz2w5ztK3zNmOoDrA)
![{\displaystyle {\vec {g}}\cdot (\mathbf {A} \times {\vec {a}}):=({\vec {g}}\cdot \mathbf {A} )\times {\vec {a}}=(\mathbf {A} ^{\top }\cdot {\vec {g}})\times {\vec {a}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzDwOntG5oNGOzqe3ntFEaghBoDsPoNs5atrDztG0aghDzDa5nDhE)
![{\displaystyle (\mathbf {A} \times {\vec {a}})\cdot {\vec {b}}=\mathbf {A} \cdot ({\vec {a}}\times {\vec {b}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81yjBCntzBytC0ngw0oAvBo2sQzgsQaNeNytK5ngw5oAw5aNnAoDK0)
![{\displaystyle {\vec {a}}\times \mathbf {A} =-\left(\mathbf {A} ^{\top }\times {\vec {a}}\right)^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ajvEzNFEaqw5aAe0ztvFntmOoNnCyjKNotJDzgiQzNdEagvFytBE)
![{\displaystyle \mathbf {A} \times {\vec {a}}=-\left({\vec {a}}\times \mathbf {A} ^{\top }\right)^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CoqzBoqiNatiNotw0yjnAaDCNatK0zqrEaAe2ytsPnDeNzts0atvB)
Symmetrische Tensoren:
Insbesondere Kugeltensoren:
Schiefsymmetrische Tensoren:
#Axialer Tensor oder Kreuzproduktmatrix mit dem #Einheitstensor:
![{\displaystyle ({\vec {a}}\times \mathbf {1} )\cdot {\vec {g}}={\vec {a}}\cdot ({\vec {g}}\times \mathbf {1} )={\vec {a}}\cdot (\mathbf {1} \times {\vec {g}})={\vec {a}}\times {\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EyjzAoDvFz2wPotFAztnFyjCQaNvCzteQa2e2oNs2ntnEytwOaNeO)
Mehrfach:
![{\displaystyle ({\vec {a}}\times ({\vec {b}}\times \mathbf {A} ))\cdot {\vec {g}}={\vec {a}}\times ({\vec {b}}\times (\mathbf {A} \cdot {\vec {g}}))=({\vec {a}}\cdot \mathbf {A} \cdot {\vec {g}}){\vec {b}}-({\vec {a}}\cdot {\vec {b}})\mathbf {A} \cdot {\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PygzEzDe4oDs2aqo4o2zByti5ntG3oqwQoDC4nqw3aAs3z2a4ztK2)
![{\displaystyle {\vec {a}}\times ({\vec {b}}\times \mathbf {A} )={\vec {b}}\otimes {\vec {a}}\cdot \mathbf {A} -({\vec {a}}\cdot {\vec {b}})\mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82ntvDaDdCygo1a2s1agdCyjsQytC4ote5zjs3oNnAoNlDaNdFajrC)
Meistens ist aber:
![{\displaystyle (\mathbf {A} \cdot {\vec {a}})\times {\vec {g}}\neq \mathbf {A} \cdot ({\vec {a}}\times {\vec {g}})=(\mathbf {A} \times {\vec {a}})\cdot {\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eo2w2ytzAoArEnqhEaje0nAe0ygs4oAoOage3ngrEato2aAw3ntK0)
![{\displaystyle {\vec {a}}\times ({\vec {g}}\cdot \mathbf {A} )\neq ({\vec {a}}\times {\vec {g}})\cdot \mathbf {A} ={\vec {a}}\cdot ({\vec {g}}\times \mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OaDdBnjm2yte5ngzBaqrFytCQaDm2yjoQzqdBaqoOoDG1a2s5otzC)
Abbildung
![{\displaystyle \mathbf {A\times B} ={\stackrel {3}{\mathbf {E} }}:(\mathbf {A\cdot B} ^{\top })=-{\stackrel {3}{\mathbf {E} }}:(\mathbf {B\cdot A} ^{\top })=-\mathbf {B\times A} \in \mathbb {V} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NaNrCz2nDaAhDnDsQajoQnAaPntK5ztaOaqrCztnEzDs5oDJEztoP)
mit #Fundamentaltensor 3. Stufe
.
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\times ({\vec {b}}\otimes {\vec {h}})=({\vec {g}}\cdot {\vec {h}}){\vec {a}}\times {\vec {b}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OaNzFnjlFyjw0zgo1aNJDaqw3njrEagdCyjw1zqdCatoOnjsQzDwO)
![{\displaystyle {\begin{aligned}&A_{ik}({\hat {e}}_{i}\otimes {\hat {e}}_{k})\times [B_{jl}({\hat {e}}_{j}\otimes {\hat {e}}_{l})]=A_{ik}B_{jk}({\hat {e}}_{i}\times {\hat {e}}_{j})=\ldots \\&\ldots ={\begin{pmatrix}A_{21}B_{31}-A_{31}B_{21}+A_{22}B_{32}-A_{32}B_{22}+A_{23}B_{33}-A_{33}B_{23}\\A_{31}B_{11}-A_{11}B_{31}+A_{32}B_{12}-A_{12}B_{32}+A_{33}B_{13}-A_{13}B_{33}\\A_{11}B_{21}-A_{21}B_{11}+A_{12}B_{22}-A_{22}B_{12}+A_{13}B_{23}-A_{23}B_{13}\end{pmatrix}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84nqi5ntnEzNG5zqvEaqw2oNK4o2e1zNzEnts2oqdCaNzCytzDnAaP)
Zusammenhang mit #Dualer axialer Vektor und #Vektorinvariante:
![{\displaystyle \mathbf {A\times B} =-2{\stackrel {A}{\overrightarrow {\mathbf {A\cdot B} ^{\top }}}}={\vec {\mathrm {i} }}(\mathbf {A\cdot B} ^{\top })}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Paqe2aNlDatdAzqa0z2zFzthFzNKPagaQaDwQots5otK2zqw4yjw1)
Mit #Einheitstensor:
![{\displaystyle \mathbf {1\times A} =2{\stackrel {A}{\overrightarrow {\mathbf {A} }}}=-{\vec {\mathrm {i} }}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84aDFAztaQzDGQaAzDyjC1ytsQnjiOnAeOzjs0oAaOnAvAzDw0aqiO)
Mehrfachprodukte:
![{\displaystyle (\mathbf {A\cdot B} )\times \mathbf {C} =\mathbf {A} \times (\mathbf {C\cdot B} ^{\top })}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85oNiOaNJAyqi5o2nEytoQaNmPnDs4zqvEatFBytG5ati4agdAaDw0)
![{\displaystyle \mathbf {A} \times (\mathbf {B\cdot C} )=(\mathbf {A\cdot C} ^{\top })\times \mathbf {B} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aAoQytFDoqs1ygsOotnCnDsNzAi5yto4nArCoAwNajdBztCQzNFA)
Zusammenhang mit dem #Skalarkreuzprodukt von Tensoren:
![{\displaystyle \mathbf {A\times B} =\mathbf {A} \cdot \!\!\times (\mathbf {B} ^{\top })}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzjJFyje3oNwOzAw0oDaPaqvAnjK5zDo5aAhCnDs2zqeOzDsOztJE)
Abbildung
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\cdot \!\!\times ({\vec {h}}\otimes {\vec {u}})=-({\vec {u}}\otimes {\vec {h}})\cdot \!\!\times ({\vec {g}}\otimes {\vec {a}}):=({\vec {g}}\cdot {\vec {h}}){\vec {a}}\times {\vec {u}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaAo5yga3zArEajGNajiOyto1ztlBzDlAaDi3zjhEzti3otw1aDiP)
![{\displaystyle {\begin{aligned}&A_{ik}({\hat {e}}_{i}\otimes {\hat {e}}_{k})\cdot \!\!\times [B_{lj}({\hat {e}}_{l}\otimes {\hat {e}}_{j})]=A_{ik}B_{kj}({\hat {e}}_{i}\times {\hat {e}}_{j})=\ldots \\&\ldots ={\begin{pmatrix}A_{21}B_{13}-A_{31}B_{12}+A_{22}B_{23}-A_{32}B_{22}+A_{23}B_{33}-A_{33}B_{32}\\A_{31}B_{11}-A_{11}B_{13}+A_{32}B_{21}-A_{12}B_{23}+A_{33}B_{31}-A_{13}B_{33}\\A_{11}B_{12}-A_{21}B_{11}+A_{12}B_{22}-A_{22}B_{21}+A_{13}B_{32}-A_{23}B_{31}\end{pmatrix}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qo2s1ngeOytnDnAiPzge2zAzDajG0ztCOz2hAoAs0ygeNytwNotKN)
Das Skalarkreuzprodukt mit dem #Einheitstensor vertauscht das dyadische Produkt durch das Kreuzprodukt:
![{\displaystyle \mathbf {1} \cdot \!\!\times ({\vec {a}}\otimes {\vec {b}})={\vec {a}}\times {\vec {b}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82atK4oAsPnjs1nqwNytm0oqvFzAoPnqhFa2hAzqaOaqs5aDwQoNrF)
Allgemein:
![{\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} =-(\mathbf {B} ^{\top })\cdot \!\!\times (\mathbf {A} ^{\top })}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81zNs2oNFDnAwNzNK4zDhEate1zDo0nDoQzDG4zqoQaNzFyta0ngo4)
![{\displaystyle \mathbf {A} \cdot \!\!\times (\mathbf {B\cdot C} )=(\mathbf {A\cdot B} )\cdot \!\!\times \mathbf {C} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzNvFoNGOaNJEzgi4yjnDoDnEzNmPajrEytGOagsOags4athDzqnE)
![{\displaystyle (\mathbf {A\cdot B} )\cdot \!\!\times \mathbf {C} =\mathbf {A} \cdot \!\!\times (\mathbf {B\cdot C} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zja3ajlFaNC4zNGPzgi4ntm3atzAaDG4ztBEzArBaDGQnjG2zAs1)
Zusammenhang mit dem #Kreuzprodukt von Tensoren:
![{\displaystyle \mathbf {S} \cdot \!\!\times \mathbf {T} =\mathbf {S\times (T^{\top })} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eyjs0oArEytKOzgo5aAnBztG4zDzEyga3oqe2otiQzNzEaAdAyjKQ)
Zusammenhang mit #Vektorinvariante und #Dualer axialer Vektor:
![{\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} ={\vec {\mathrm {i} }}(\mathbf {A} \cdot \mathbf {B} )=-2{\stackrel {A}{\overrightarrow {\mathbf {A} \cdot \mathbf {B} }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PzDG0ntBEoDG5oDsQnqzDyqvEnDhAoNdCzjFDnjo4o2wPaqhEa2dE)
Siehe auch #Äußeres Tensorprodukt #
Abbildung
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\times \!\!\times ({\vec {h}}\otimes {\vec {b}}):=({\vec {g}}\times {\vec {h}})\otimes ({\vec {a}}\times {\vec {b}})=({\vec {g}}\otimes {\vec {a}})\#({\vec {h}}\otimes {\vec {b}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CotK3zDBFaqs4oqe4zje2ajJBnAaOyqo0nDaNaDsQytFEaNe2aDw4)
![{\displaystyle A_{ij}({\hat {e}}_{i}\otimes {\hat {e}}_{j})\times \!\!\times [B_{kl}({\hat {e}}_{k}\otimes {\hat {e}}_{l})]:=A_{ij}B_{kl}({\hat {e}}_{j}\times {\hat {e}}_{k})\otimes ({\hat {e}}_{i}\times {\hat {e}}_{l})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85oDm0othCzqrDnjJCytrEyjoQzgsNzjaOyjK3zNhFnghEnAe3ajCN)
![{\displaystyle \mathbf {A} \times \!\!\times \mathbf {B} =\mathbf {A} ^{\top }\#\mathbf {B} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PotdByjCOoNvAzjG5aqhAnDaNaNK0aDdCzDo0zjw2oNi5ati0o2a4)
Abbildung
![{\displaystyle ({\vec {a}}\otimes {\vec {g}})\#({\vec {b}}\otimes {\vec {h}}):=({\vec {a}}\times {\vec {b}})\otimes ({\vec {g}}\times {\vec {h}})=({\vec {g}}\otimes {\vec {a}})\times \!\!\times ({\vec {b}}\otimes {\vec {h}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OaAdBzgs2aDGPatBAnqe2otvDzNFAaNm4oqe5ztC5aDlFoNdBygi4)
![{\displaystyle {\begin{aligned}&(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})\#(B_{kl}{\hat {e}}_{k}\otimes {\hat {e}}_{l})=A_{ij}B_{kl}({\hat {e}}_{i}\times {\hat {e}}_{k})\otimes ({\hat {e}}_{j}\times {\hat {e}}_{l})\\&\qquad \qquad \qquad \qquad \qquad \quad \;\;\;=\epsilon _{ikm}\epsilon _{jln}A_{ij}B_{kl}{\hat {e}}_{m}\otimes {\hat {e}}_{n}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DathAzjlCaDC5oDsNaDo5zto1aAaNoDGQaNo5aqe4ytrAzjo2oqo1)
Mit der Formel für das Produkt zweier #Permutationssymbole:
![{\displaystyle {\begin{aligned}\mathbf {A} \#\mathbf {B} =&[\mathrm {Sp} (\mathbf {A} )\mathrm {Sp} (\mathbf {B} )-\mathrm {Sp} (\mathbf {A\cdot B} )]\mathbf {1} \\&+[\mathbf {A\cdot B} +\mathbf {B\cdot A} -\mathrm {Sp} (\mathbf {A} )\mathbf {B} -\mathrm {Sp} (\mathbf {B} )\mathbf {A} ]^{\top }\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AaNlFots2nta5ajlFnte4otvFygsPztK0njKNnjm1ngiOaNnDzAaN)
Grundlegende Eigenschaften:
![{\displaystyle \mathbf {A} \#\mathbf {B} =\mathbf {B} \#\mathbf {A} =(\mathbf {A} ^{\top }\#\mathbf {B} ^{\top })^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzjzCyqo5ngzCatm1oti0nDzEzjw3yjwOnjvEzDm1z2o5ytm4ntvF)
![{\displaystyle (\mathbf {A+B} )\#\mathbf {C} =\mathbf {A} \#\mathbf {C} +\mathbf {B} \#\mathbf {C} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aqdEnAa4njzAnts0nDs1a2sOaNsOztGPaji3yqhCaAiNzgo3zjBE)
![{\displaystyle \mathbf {A} \#(\mathbf {B+C} )=\mathbf {A} \#\mathbf {B} +\mathbf {A} \#\mathbf {C} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DnDvCyqzDnDKNyte1oDaNygwQatBAoDeOagnCnjaPags3nDnAa2o0)
Kreuzprodukt und #Kofaktor:
![{\displaystyle (\mathbf {A} \#\mathbf {B} )\cdot ({\vec {u}}\times {\vec {v}})=(\mathbf {A} \cdot {\vec {u}})\times (\mathbf {B} \cdot {\vec {v}})-(\mathbf {A} \cdot {\vec {v}})\times (\mathbf {B} \cdot {\vec {u}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82yje1njdDztBEzAsPajnFaArByjaPoDKOaNlFyjdBotlEngs3zNlC)
![{\displaystyle {\frac {1}{2}}(\mathbf {A} \#\mathbf {A} )\cdot ({\vec {u}}\times {\vec {v}})=\mathrm {cof} (\mathbf {A} )\cdot ({\vec {u}}\times {\vec {v}})=(\mathbf {A} \cdot {\vec {u}})\times (\mathbf {A} \cdot {\vec {v}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CntmNnqdDotKOz2i3aje3aDa0aNm1ajvBoDG0a2eQzDe3nqeNoDKP)
#Hauptinvarianten:
![{\displaystyle {\frac {1}{2}}(\mathbf {A\#1} ):\mathbf {1} =\mathrm {Sp} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DyqnAztwNajBFaDs4yjG1aNa1oAw2ajm4ytG1ntJFnjnDntvCzNC2)
![{\displaystyle {\frac {1}{2}}(\mathbf {A\#A} ):\mathbf {1} =\mathrm {I} _{2}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QoDeQztJFotmOzAaQoAnDnjs3atG3ntBBntCNo2vBnqnEzNlDaNhD)
![{\displaystyle {\frac {1}{6}}(\mathbf {A\#A} ):\mathbf {A} =\det(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FzgnDytwPzgs0oDJCothEzgvAzNiNzDiOoDe0nAvDajmOzDnDzDw4)
Weitere Eigenschaften:
![{\displaystyle \mathbf {1} \#\mathbf {1} =2\,\mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OyqzAztK1zNnEzqhAyqa2o2hCzNmPoAvCnjaNo2dEnAi3nDzDoqzB)
![{\displaystyle \mathbf {A} \#\mathbf {1} =\mathrm {Sp} (\mathbf {A} )\mathbf {1} -\mathbf {A} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FaNw0oDoNzDdEnjrBnAoNytCPnAa0zDe4ytm1a2zEztvEzjJEzNaO)
![{\displaystyle (\mathbf {A} \#\mathbf {B} ):\mathbf {C} =(\mathbf {B} \#\mathbf {C} ):\mathbf {A} =(\mathbf {C} \#\mathbf {A} ):\mathbf {B} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PzNm1z2dFoDoQoDsQnqrBnjs0nDsOntFDoqhFotG4nghDaDJDzjC1)
![{\displaystyle \mathrm {Sp} (\mathbf {A} \#\mathbf {B} )=\mathrm {Sp} (\mathbf {A} )\mathrm {Sp} (\mathbf {B} )-\mathrm {Sp} (\mathbf {A\cdot B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dztw5oNGOntmPngzDngs4nqdFnjCQnje5njo2aNhFnqaNaDzAoDa3)
![{\displaystyle (\mathbf {A} \#\mathbf {B} )\cdot (\mathbf {C} \#\mathbf {D} )=(\mathbf {A\cdot C} )\#(\mathbf {B\cdot D} )+(\mathbf {A\cdot D} )\#(\mathbf {B\cdot C} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80yjFEyteOoNdDaqo0agvBnjCPyjzDzDK0nqi5oAoNaji1nDnCnAi5)
Aber meistens:
![{\displaystyle (\mathbf {A} \#\mathbf {B} )\#\mathbf {C} \neq \mathbf {A} \#(\mathbf {B} \#\mathbf {C} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ngzFzgzBzqi2ygw3yja5zDG1nAaPzDm4ygnFzqnAngrCzjvCo2e2)
.
![{\displaystyle \mathbf {A} \cdot ({\vec {a}}\otimes {\vec {g}})=(\mathbf {A} \cdot {\vec {a}})\otimes {\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81nDwOotFBzAnDngdBnjC5atwPzthBnAe5zNsQnAwPoDhEngzAnjo4)
![{\displaystyle {\vec {a}}\otimes (\mathbf {A} \cdot {\vec {g}})=({\vec {a}}\otimes {\vec {g}})\cdot \mathbf {A} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80oDmOotC3aAs1aAs0ygrDngsPote4oDnBzAi5oDJFo2s1njG4a2wQ)
![{\displaystyle {\vec {a}}\cdot \mathbf {A} \cdot {\vec {g}}=\mathbf {A} :({\vec {a}}\otimes {\vec {g}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NajdAzteNztdDnAdEoAeNzNlAaNaNoDaOotnDatC1ntK4nDK5o2rD)
Spatprodukt und #Determinante eines Tensors:
![{\displaystyle (\mathbf {A} \cdot {\vec {a}})\cdot [(\mathbf {A} \cdot {\vec {b}})\times (\mathbf {A} \cdot {\vec {c}})]=\mathrm {det} (\mathbf {A} )\;{\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DoAa3oNw5oDzBoDFAnDJDyqhEaNJEnjK3aja4ytsNajnEz2hFngsP)
Kreuzprodukt und #Kofaktor:
![{\displaystyle (\mathbf {A} \cdot {\vec {a}})\times (\mathbf {A} \cdot {\vec {b}})=\mathrm {cof} (\mathbf {A} )\cdot ({\vec {a}}\times {\vec {b}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BatrEnqi4a2wOnjlCo2w2nDJEyjC2njJEnDFBa2vDoNnAzjhAzjhF)
![{\displaystyle \mathbf {A} ^{\top }\cdot [(\mathbf {A} \cdot {\vec {a}})\times (\mathbf {A} \cdot {\vec {b}})]=\mathrm {det} (\mathbf {A} )\;{\vec {a}}\times {\vec {b}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnDzDzDoOzqaQo2oNoAzCzjm2otCNoNvEzNiNotmNota1zNmQnDvF)
#Axialer Tensor oder Kreuzproduktmatrix, #Kreuzprodukt von Tensoren, #Skalarkreuzprodukt von Tensoren, #Dualer axialer Vektor und #Vektorinvariante:
![{\displaystyle ({\vec {u}}\times \mathbf {1} )\cdot {\vec {v}}=({\vec {u}}\otimes {\vec {v}})\times \mathbf {1} =({\vec {u}}\otimes {\vec {v}})\cdot \!\!\times \mathbf {1} ={\stackrel {A}{\overrightarrow {({\vec {u}}\times {\vec {v}})\times \mathbf {1} }}}={\vec {\mathrm {i} }}({\vec {u}}\otimes {\vec {v}})={\vec {u}}\times {\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zqrAoDaOyjdBntm3njoNzgwNnAhEotKNoAo1yga0otmOz2eOntzE)
![{\displaystyle \mathbf {A} =A_{ij}\,{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{pmatrix}}\quad \rightarrow \;A_{ij}={\hat {e}}_{i}\cdot \mathbf {A} \cdot {\hat {e}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81aNvEatm2zDdFzAzEnjm3oNoNygwQnjKQo2w0njJFaAa3aDaNzga3)
![{\displaystyle \mathbf {A} =A^{ij}\,{\vec {a}}_{i}\otimes {\vec {g}}_{j}\quad \rightarrow \;A^{ij}={\vec {a}}^{i}\cdot \mathbf {A} \cdot {\vec {g}}^{j}=({\vec {a}}^{i}\otimes {\vec {g}}^{j}):\mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AyqaPzDo1aqhEzjvFzqzBz2eOajBBzthBnDFEzjG5zNG1ajoOnqvE)
![{\displaystyle \mathbf {A} =A_{ij}\,{\vec {a}}^{i}\otimes {\vec {g}}^{j}\quad \rightarrow \;A_{ij}={\vec {a}}_{i}\cdot \mathbf {A} \cdot {\vec {g}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PzqsOaDBAytmNzqa5nqw2a2w5nDsNnjm5nDi1yqs5zNiQotw3ytsN)
![{\displaystyle \mathbf {A} =A_{j}^{i}\,{\vec {a}}_{i}\otimes {\vec {g}}^{j}\quad \rightarrow \;A_{j}^{i}={\vec {a}}^{i}\cdot \mathbf {A} \cdot {\vec {g}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84nDhAyqw0oqw4aDsOnqaQnjmPzjKPoAeOzjK1nto2ztnCoqs0atFF)
![{\displaystyle \mathbf {A} =A_{i}^{j}\,{\vec {a}}^{i}\otimes {\vec {g}}_{j}\quad \rightarrow \;A_{i}^{j}={\vec {a}}_{i}\cdot \mathbf {A} \cdot {\vec {g}}^{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85a2o1ajm3aNrDzqdFnje2ntmPoti0o2ePzAsOnDBFztdFajCQajGQ)
![{\displaystyle \mathbf {A} =A_{ij}{\vec {a}}^{i}\otimes {\vec {a}}^{j}=A_{ij}^{\ast }{\vec {b}}^{i}\otimes {\vec {b}}^{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnjlCnqa4ntwOajm5zAhFntC0otnEzjiOoNCPyqrCaAnAytm3atJA)
Die Komponenten
ergeben sich durch Vor- und Nachmultiplikation mit dem #Einheitstensor
:
![{\displaystyle {\begin{aligned}\mathbf {A} =\mathbf {1\cdot A\cdot 1} ^{\top }=&({\vec {b}}^{i}\otimes {\vec {b}}_{i})\cdot (A_{kl}{\vec {a}}^{k}\otimes {\vec {a}}^{l})\cdot ({\vec {b}}_{j}\otimes {\vec {b}}^{j})\\=&({\vec {b}}_{i}\cdot {\vec {a}}^{k})A_{kl}({\vec {a}}^{l}\cdot {\vec {b}}_{j}){\vec {b}}^{i}\otimes {\vec {b}}^{j}=:A_{ij}^{\ast }{\vec {b}}^{i}\otimes {\vec {b}}^{j}\\\rightarrow A_{ij}^{\ast }=&({\vec {b}}_{i}\cdot {\vec {a}}^{k})A_{kl}({\vec {a}}^{l}\cdot {\vec {b}}_{j})\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzDm3yjJBnta5ajmPygsNzNKOaNnDnjs1aNzFaAwNoAdAa2a1zAeQ)
Allgemein:
![{\displaystyle \mathbf {A} =A_{ij}{\vec {a}}^{i}\otimes {\vec {g}}^{j}=A_{ij}^{\ast }{\vec {b}}^{i}\otimes {\vec {h}}^{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zDK2njs3ngw3aqvAotKPa2rAnAwQaqsQagzDygaNa2i1zgiOz2dE)
Basiswechsel mit
:
![{\displaystyle {\begin{aligned}\mathbf {A} =\mathbf {1\cdot A\cdot 1} ^{\top }=&({\vec {b}}_{i}\cdot {\vec {a}}^{k})({\vec {b}}^{i}\otimes {\vec {a}}_{k})\cdot A_{mn}({\vec {a}}^{m}\otimes {\vec {g}}^{n})\cdot ({\vec {h}}_{j}\cdot {\vec {g}}^{l})({\vec {g}}_{l}\otimes {\vec {h}}^{j})\\=&({\vec {b}}_{i}\cdot {\vec {a}}^{k})A_{kl}({\vec {h}}_{j}\cdot {\vec {g}}^{l})({\vec {b}}^{i}\otimes {\vec {h}}^{j})=A_{ij}^{\ast }{\vec {b}}^{i}\otimes {\vec {h}}^{j}\\\rightarrow A_{ij}^{\ast }=&({\vec {b}}_{i}\cdot {\vec {a}}^{k})A_{kl}({\vec {g}}^{l}\cdot {\vec {h}}_{j})\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OaAdAnAdFz2zAate2oqwQygi3nti3oNK2aAi1aDs4nqzFnAs1zNwO)
Definition für einen Tensor A:
![{\displaystyle \langle {\vec {u}},{\vec {v}}\rangle :={\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}=\mathbf {A} :({\vec {u}}\otimes {\vec {v}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnDBEatvAagiPoNmNzqaPagnDaDK3zjhEz2dFoqhAzNG4oto0nDaN)
Zwei Tensoren A und B sind identisch, wenn
![{\displaystyle {\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}={\vec {u}}\cdot \mathbf {B} \cdot {\vec {v}}\quad \forall \;{\vec {u}},{\vec {v}}\in \mathbb {V} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QotFBzjC3othDnji0agnDoqvBaqnFntdDajoNzteNagnByjo0a2s5)
Definition
![{\displaystyle \mathrm {cof} (\mathbf {A} ):=\mathbf {A^{\top }\cdot A^{\top }} -\mathrm {I} _{1}(\mathbf {A} )\mathbf {A} ^{\top }+\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PztmPajhDyjnDaNC4ytFCati5nAoQatzEoAwQnjhEnjdBygw2zqiQ)
#Invarianten:
Wenn λ1,2,3 die #Eigenwerte des Tensors A sind, dann hat cof(A) die Eigenwerte λ1λ2, λ2λ3, λ3λ1.
#Hauptinvarianten:
![{\displaystyle \mathrm {I} _{1}(\mathrm {cof} (\mathbf {A} ))=\mathrm {I} _{2}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ento5aNG3aDaQa2vAzjFBzDm2oDFCygi0zNFAzDvCnDJEz2ePzjhB)
![{\displaystyle \mathrm {I} _{2}(\mathrm {cof} (\mathbf {A} ))=\mathrm {I} _{1}(\mathbf {A} )\mathrm {det} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ngvCztGNngnDytCPagiPzgsNzjnFoDs1nDs4ytGPoNs4nAzEzjiO)
![{\displaystyle \mathrm {det} (\mathrm {cof} (\mathbf {A} ))=\mathrm {det} ^{2}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81aArFzghBzjnEztnCataPoNG2oqa1oqrDo2nBotw2ajCOote3nDw0)
#Betrag:
![{\displaystyle \|\mathrm {cof} (\mathbf {A} )\|={\sqrt {\mathrm {I} _{2}(\mathbf {A^{\top }\cdot A} )}}={\frac {\sqrt {2}}{2}}{\sqrt {\|\mathbf {A} \|^{4}-\|\mathbf {A^{\top }\cdot A} \|^{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NaDnFyjzFotC5nqdDoDC2z2hCoqdAnDlAnDaNyqiPoqa3ate5agrE)
Weitere Eigenschaften:
![{\displaystyle \mathrm {cof} (x\mathbf {A} )=x^{2}\mathrm {cof} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fo2i4aNo4ygi0ytKOzgsQzgzDyqiPzDa4aAhAygiQzAwPytGNaqeP)
![{\displaystyle \mathrm {det} (\mathbf {A} )\neq 0\quad \rightarrow \quad \mathrm {cof} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {A} ^{\top -1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81yjlBntnCyqdEygo4atoNaAhDzqo5yqvBzNsQnDBDz2vByji3ajwP)
![{\displaystyle \mathbf {A} ^{\top }\cdot \mathrm {cof} (\mathbf {A} )=\mathrm {cof} (\mathbf {A} )\cdot \mathbf {A} ^{\top }=\mathrm {det} (\mathbf {A} )\mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PoNdBoNrEoqhFatGOaDK5zAaQz2wPzNiQo2a0yjePaDeOzteOzgw0)
![{\displaystyle \mathrm {cof} (\mathbf {A\cdot B} )=\mathrm {cof} (\mathbf {A} )\cdot \mathrm {cof} (\mathbf {B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QotK5zgnAzNFDzNi0otBDatKPyteNoAaQzjFFatdBzjGPotBFaDhC)
![{\displaystyle \mathrm {cof} (\mathbf {A} ^{\top })=\mathrm {cof} (\mathbf {A} )^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OaNKOngePzNmNnDrEyjrAyto3aqsOaDw3ztmOoNw2oAnDoto2zAzD)
![{\displaystyle \mathrm {cof} \left(\mathrm {cof} (\mathbf {A} )\right)=\mathrm {det} (\mathbf {A} )\mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FoNiOzDhFygwOoNGPytzBygwQoNJEatBEaAs5yjdBngo2zgvAytrF)
![{\displaystyle {\begin{aligned}&\mathrm {cof} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})={\frac {1}{2}}(A_{kl}A_{mn}\epsilon _{kmi}\epsilon _{lnj})({\hat {e}}_{i}\otimes {\hat {e}}_{j})=\ldots \\&\ldots ={\begin{pmatrix}A_{22}A_{33}-A_{23}A_{32}&A_{23}A_{31}-A_{21}A_{33}&A_{21}A_{32}-A_{22}A_{31}\\A_{32}A_{13}-A_{33}A_{12}&A_{33}A_{11}-A_{31}A_{13}&A_{31}A_{12}-A_{32}A_{11}\\A_{12}A_{23}-A_{13}A_{22}&A_{13}A_{21}-A_{11}A_{23}&A_{11}A_{22}-A_{12}A_{21}\end{pmatrix}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnjzCa2e0othAaNJByqvCnDa0ztFFaNs4aNzFaNa3a2w3zjFEztJA)
Kofaktor und #Äußeres Tensorprodukt:
![{\displaystyle \mathrm {cof} (\mathbf {A} )={\frac {1}{2}}\mathbf {A} \#\mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zjo2ntK5aNvCnqrAyqw2zDvDzDs0otK4oNG0ytJFzAs4zNi2ntBA)
![{\displaystyle {\begin{aligned}\mathrm {cof} (\mathbf {A+B} )=&{\frac {1}{2}}(\mathbf {A} \#\mathbf {A} +2\mathbf {A} \#\mathbf {B} +\mathbf {B} \#\mathbf {B} )\\=&\mathrm {cof} (\mathbf {A} )+\mathrm {cof} (\mathbf {B} )+\mathbf {A} \#\mathbf {B} \end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QoNC0zqnEzga2ajBDnAo3agnCnqe0atJEyqs4yjzBotK4aAw3aqa4)
Kreuzprodukt und Kofaktor:
![{\displaystyle (\mathbf {A} \cdot {\vec {a}})\times (\mathbf {A} \cdot {\vec {b}})=\mathrm {cof} (\mathbf {A} )\cdot ({\vec {a}}\times {\vec {b}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BatrEnqi4a2wOnjlCo2w2nDJEyjC2njJEnDFBa2vDoNnAzjhAzjhF)
Definition:
![{\displaystyle \mathrm {adj} (\mathbf {A} ):=\mathbf {A} ^{2}-\mathrm {I} _{1}(\mathbf {A} )\mathbf {A} +\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} =\mathrm {cof} (\mathbf {A} )^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aDeQnDJFngsQaDG3yge3nDa0zghBaqwQoNa5ajnAoDo3o2s1zNeQ)
#Hauptinvarianten:
![{\displaystyle \mathrm {I} _{1}(\mathrm {adj} (\mathbf {A} ))=\mathrm {I} _{2}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzNKQoAvCzqzFytmOajvCotG2nje1zDsNyjBFzAa4oAeQaNJBytm2)
![{\displaystyle \mathrm {I} _{2}(\mathrm {adj} (\mathbf {A} ))=\mathrm {I} _{1}(\mathbf {A} )\mathrm {det} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85ntiQnqzAnghDajKOote0oDlDzji5nDzAaDlCyqvFajCPyjBFoDBA)
![{\displaystyle \mathrm {det} (\mathrm {adj} (\mathbf {A} ))=\mathrm {det} ^{2}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BzDBEztnAaDlEajlFoNFFz2vAz2nBz2zCoAzBaqa2aqaPoAs2oNnF)
#Betrag:
![{\displaystyle \|\mathrm {adj} (\mathbf {A} )\|={\sqrt {\mathrm {I} _{2}(\mathbf {A^{\top }\cdot A} )}}={\frac {\sqrt {2}}{2}}{\sqrt {\|\mathbf {A} \|^{4}-\|\mathbf {A^{\top }\cdot A} \|^{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zNe3oNoQnAdEnjzEz2w1zNBAz2e3zqnAzAzAnAw4aDG0aAa2aNCP)
Weitere Eigenschaften:
![{\displaystyle \mathrm {adj} (x\mathbf {A} )=x^{2}\mathrm {adj} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81agoPztJFyqo1aDdAnqe3yqnAotGQaAo2ajJEzDhEzAo2ajePnts3)
![{\displaystyle \mathrm {det} (\mathbf {A} )\neq 0\quad \rightarrow \quad \mathrm {adj} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {A} ^{-1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaDhAnje1yts1aDi4oDs5zjvEaAaNnDvBytKNzNo2zDG2zNK5zqrF)
![{\displaystyle \mathbf {A} \cdot \mathrm {adj} (\mathbf {A} )=\mathrm {adj} (\mathbf {A} )\cdot \mathbf {A} =\mathrm {det} (\mathbf {A} )\mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzDe5ngw0o2w1aAnDygvEnDo5zjm0ytC4yjsOzqiPajlDzNBCzDvE)
![{\displaystyle \mathrm {adj} (\mathbf {A\cdot B} )=\mathrm {adj} (\mathbf {B} )\cdot \mathrm {adj} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzNe1oNaNnDiPnti5aAaOaNaNntwQaAo3agvDo2vDnDG2agi3oto2)
![{\displaystyle \mathrm {adj} (\mathbf {A} ^{\top })=\mathrm {adj} (\mathbf {A} )^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PotiOa2wOotG5oqs0oteNoArCoNrEyjnDzDC0nqnAajiOatC4zjnB)
![{\displaystyle {\begin{aligned}\mathrm {adj} (\mathbf {A+B} )=&{\frac {1}{2}}(\mathbf {A} \#\mathbf {A} +2\mathbf {A} \#\mathbf {B} +\mathbf {B} \#\mathbf {B} )^{\top }\\=&\mathrm {adj} (\mathbf {A} )+\mathrm {adj} (\mathbf {B} )+\mathbf {A} ^{\top }\#\mathbf {B} ^{\top }\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AaqsNyjwNoNi3oDa1aqs2oAhBzAi3z2iOyta3njmOnAoPztKQntGQ)
![{\displaystyle \mathrm {adj} \left(\mathrm {adj} (\mathbf {A} )\right)=\mathrm {det} (\mathbf {A} )\mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85nDG2otsNntzAoDJDzji4njhEaAs0zNoOzqnFzjdBzqwPoqrFoNi5)
![{\displaystyle {\begin{aligned}&\mathrm {adj} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})={\frac {1}{2}}(A_{kl}A_{mn}\epsilon _{kmj}\epsilon _{lni})({\hat {e}}_{i}\otimes {\hat {e}}_{j})=\ldots \\&\ldots ={\begin{pmatrix}A_{22}A_{33}-A_{23}A_{32}&A_{32}A_{13}-A_{33}A_{12}&A_{12}A_{23}-A_{13}A_{22}\\A_{23}A_{31}-A_{21}A_{33}&A_{33}A_{11}-A_{31}A_{13}&A_{13}A_{21}-A_{11}A_{23}\\A_{21}A_{32}-A_{22}A_{31}&A_{31}A_{12}-A_{32}A_{11}&A_{11}A_{22}-A_{12}A_{21}\end{pmatrix}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzDdEajKPyjCPnqvEzNG2aDlAyjiNytC4aga0a2e1ygnAz2e1nDJD)
Definition
![{\displaystyle \mathbf {A} ^{-1}:\quad \mathbf {A} ^{-1}\cdot \mathbf {A} =\mathbf {A\cdot A} ^{-1}=\mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zjdCzgwQajaNoNsPyto1nAiNzAaPoAaQotvDnqe3atFEoDGQzDw0)
Die Inverse ist nur definiert, wenn
Zusammenhang mit dem adjungierten Tensor
:
![{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\det(\mathbf {A} )}}\mathrm {adj} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81a2o1njCPoNe5zjzCyqzBzDFFo2dBz2w5ztJFoAsNngzAo2zAntJD)
![{\displaystyle {\begin{aligned}\mathbf {A} =&A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})\\\rightarrow \mathbf {A} ^{-1}=&{\frac {1}{|\mathbf {A} |}}{\begin{pmatrix}A_{22}A_{33}-A_{23}A_{32}&A_{32}A_{13}-A_{33}A_{12}&A_{12}A_{23}-A_{13}A_{22}\\A_{23}A_{31}-A_{21}A_{33}&A_{33}A_{11}-A_{31}A_{13}&A_{13}A_{21}-A_{11}A_{23}\\A_{21}A_{32}-A_{22}A_{31}&A_{31}A_{12}-A_{32}A_{11}&A_{11}A_{22}-A_{12}A_{21}\end{pmatrix}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PztrFati3zDdBntm1oqwOaji4ajiNnAdFntw3aqnDaDiQyje5zDwO)
Werden die Spalten von A mit Vektoren bezeichnet, also
, dann gilt:
![{\displaystyle \mathbf {A} ^{-1}={\begin{pmatrix}{\vec {a}}^{1}&{\vec {a}}^{2}&{\vec {a}}^{3}\end{pmatrix}}^{\top }={\frac {1}{\mathrm {det} (\mathbf {A} )}}{\begin{pmatrix}{\vec {a}}_{2}\times {\vec {a}}_{3}&{\vec {a}}_{3}\times {\vec {a}}_{1}&{\vec {a}}_{1}\times {\vec {a}}_{2}\end{pmatrix}}^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CoDlFnDmQzAhEathAatnEatKQoDo3ngs0oNKPoDvCoNw1ngzEygdB)
Satz von Cayley-Hamilton:
![{\displaystyle \mathbf {A} ^{-1}={\frac {1}{\mathrm {I} _{3}(\mathbf {A} )}}(\mathbf {A} ^{2}-\mathrm {I} _{1}(\mathbf {A} )\mathbf {A} +\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QyjzBztK2ygw5zDvCajmNoNa4nqdBz2i3zNG5ytCNnts5aji4yjG2)
worin
die drei #Hauptinvarianten sind.
Inverse des transponierten Tensors:
![{\displaystyle (\mathbf {A} ^{\top })^{-1}=(\mathbf {A} ^{-1})^{\top }=\mathbf {A} ^{\top -1}=\mathbf {A} ^{-\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AyjzFotlDyjFBaqa4aghDzNs4aDGOoNGNnti2zDe5a2eNnqhDytK5)
Inverse eines Tensorprodukts:
![{\displaystyle (\mathbf {A\cdot B} )^{-1}=\mathbf {B} ^{-1}\cdot \mathbf {A} ^{-1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Aytw5nDw5zNw5ajC0yjJBzjrDzAs3zAo1njvCyqzFnqvEngaQaNG2)
![{\displaystyle (x\mathbf {A} )^{-1}={\frac {1}{x}}\mathbf {A} ^{-1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dyto0aNJEzDs2ntFDaDBBzqiQnja3aghFzjdBytG3nqa4ajwOo2wP)
#Äußeres Tensorprodukt und Inverse einer Summe:
![{\displaystyle (\mathbf {A+B} )^{-1}={\frac {1}{\det(\mathbf {A+B} )}}\left(\mathrm {adj} (\mathbf {A} )+\mathrm {adj} (\mathbf {B} )+(\mathbf {A} \#\mathbf {B} )^{\top }\right)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8No2iOoqhBo2ePotm1zqwQz2iPoAdCyjK5aDrAaNK1aDCNoqrFzti4)
Invertierungsformeln:
![{\displaystyle (a\mathbf {1} +{\vec {b}}\otimes {\vec {c}})^{-1}={\frac {1}{a}}\left(\mathbf {1} -{\frac {1}{a+{\vec {b}}\cdot {\vec {c}}}}{\vec {b}}\otimes {\vec {c}}\right)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CntmOzDi2ztG5ntCQaDKOntwPnqs0aNJEaDsPz2i4zjBByga0ygvF)
![{\displaystyle {\begin{aligned}&(a\mathbf {1} +{\vec {b}}\otimes {\vec {c}}+{\vec {d}}\otimes {\vec {e}})^{-1}={\frac {1}{aD}}\left(D\mathbf {1} +{\vec {b}}\otimes (q{\vec {c}}+r{\vec {e}})+{\vec {d}}\otimes (s{\vec {c}}+t{\vec {e}})\right)\\&\qquad q=a+{\vec {d}}\cdot {\vec {e}},\quad r=-{\vec {c}}\cdot {\vec {d}},\quad s=-{\vec {b}}\cdot {\vec {e}},\quad t=a+{\vec {b}}\cdot {\vec {c}}\\&\qquad D=rs-qt\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qo2i3ytBCzte5zDzAnqsPo2i0oAw5zDmNzqe1atlAzjo4nghFoAnE)
![{\displaystyle ({\vec {a}}_{i}\otimes {\vec {g}}_{i})^{-1}={\vec {g}}^{i}\otimes {\vec {a}}^{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82nte0agdAaNK0zAdFaAhBoNi1ngaPyqnCzAzCyqi0njsNzDhDzNa0)
![{\displaystyle \mathbf {A} \cdot {\hat {v}}=\lambda {\hat {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zNm3nqsPajG1oNwPnqe4nDG2nti3otC5nDvEzjnBzDJDaNlAagi2)
mit Eigenwert
und Eigenvektor
. Die Eigenvektoren werden auf die Länge eins normiert.
Jeder Tensor hat drei Eigenwerte und drei dazugehörige Eigenvektoren. Mindestens ein Eigenwert und Eigenvektor sind reell. Die beiden anderen Eigenwerte und -vektoren können reell oder komplex sein.
Charakteristische Gleichung
![{\displaystyle \mathrm {det} (\mathbf {A} -\lambda _{i}\mathbf {1} )=-\lambda _{i}^{3}+\mathrm {I} _{1}(\mathbf {A} )\lambda _{i}^{2}-\mathrm {I} _{2}(\mathbf {A} )\lambda _{i}+\mathrm {I} _{3}(\mathbf {A} )=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83z2oPoDiOzAs2othAytoQoqnDaAzFzjCPyjhCaAdFajmNnghEz2hE)
Lösung siehe Cardanische Formeln. Die Koeffizienten sind die #Hauptinvarianten :
![{\displaystyle \mathrm {I} _{1}(\mathbf {A} ):=\mathrm {Sp} (\mathbf {A} )=\lambda _{1}+\lambda _{2}+\lambda _{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzNa3zDdBygs5agw1nqvAa2zDzqhDzgs1zjGNaDFAaDJEzAdEathC)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} ):={\frac {1}{2}}[\mathrm {I} _{1}(\mathbf {A} )^{2}-\mathrm {I} _{1}(\mathbf {A} ^{2})]=\lambda _{1}\lambda _{2}+\lambda _{2}\lambda _{3}+\lambda _{3}\lambda _{1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NoDe0otaPnAw4ajeNnjzAzAi1oqnAaji1otmNaAvCyqa5oqs5aqdC)
![{\displaystyle \mathrm {I} _{3}(\mathbf {A} ):=\mathrm {det} (\mathbf {A} )=\lambda _{1}\lambda _{2}\lambda _{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81aAhBotlEzgdByghFntm0oAwQzgiQyjnAoNzFzNm2oAdCzgrDnjhC)
Eigenvektoren
sind nur bis auf einen Faktor ≠ 0 bestimmt. Der Nullvektor ist kein Eigenvektor.
Bestimmungsgleichung:
Tensor
:
![{\displaystyle {\begin{pmatrix}A_{11}-\lambda &A_{12}&A_{13}\\A_{21}&A_{22}-\lambda &A_{23}\\A_{31}&A_{32}&A_{33}-\lambda \end{pmatrix}}\cdot {\begin{pmatrix}v_{1}\\v_{2}\\v_{3}\end{pmatrix}}={\begin{pmatrix}0\\0\\0\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnqrBaji5zDrFajBDaNBFzjK0a2wPatFAoDzEaDw2zDlFyga4ngw2)
Bestimmung mit gegebenem/angenommenem
:
![{\displaystyle {\begin{pmatrix}A_{12}&A_{13}\\A_{22}-\lambda &A_{23}\\A_{32}&A_{33}-\lambda \end{pmatrix}}\cdot {\begin{pmatrix}v_{2}\\v_{3}\end{pmatrix}}=v_{1}{\begin{pmatrix}\lambda -A_{11}\\-A_{21}\\-A_{31}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zAoOagiOzDC3ajhDztKOagvFyjw5yqhFyqe0z2e3ygi4nti2a2aP)
Geometrische Vielfachheit 1:
![{\displaystyle v_{2}=v_{1}{\frac {(\lambda -A_{33})A_{21}+A_{23}A_{31}}{(A_{22}-\lambda )(A_{33}-\lambda )-A_{23}A_{32}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EzNiPnjrBnAaQoDeNyge4ajaQngsOztFBoqeNo2rAaDwOoDnCngo0)
![{\displaystyle v_{3}=v_{1}{\frac {(\lambda -A_{22})A_{31}+A_{32}A_{21}}{(A_{22}-\lambda )(A_{33}-\lambda )-A_{23}A_{32}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zDiPo2s2zjo5atCPo2o2oDo4nthEnqeOzjvDyqiOzNFDztm0yqiP)
Geometrische Vielfachheit 2:
![{\displaystyle {\begin{pmatrix}A_{13}\\A_{23}\\A_{33}-\lambda \end{pmatrix}}v_{3}=-v_{1}{\begin{pmatrix}A_{11}-\lambda \\A_{21}\\A_{31}\end{pmatrix}}-v_{2}{\begin{pmatrix}A_{12}\\A_{22}-\lambda \\A_{32}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81atBDoqaNntm3ygi2zjwPaNG5z2dDzji0zAzAztBAztnCaAzBzAiQ)
Die Formeln bleiben richtig, wenn die Indizes {1,2,3} zyklisch vertauscht werden.
Symmetrischen Tensoren: Für das Betragsquadrat der Komponenten
der auf Betrag 1 normierten Eigenvektoren
des (komplexen) Tensors
gilt mit dessen Eigenwerten
und den Eigenwerten
der Hauptuntermatrizen von
:[1]
![{\displaystyle |v_{ij}|^{2}\prod _{k=1;k\neq i}^{n}{\big (}\lambda _{i}-\lambda _{k}{\big )}=\prod _{k=1}^{n-1}{\big (}\lambda _{i}-\mu _{jk}{\big )}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qaqo3otwOzAvBatlCzArDoDaQotrBaAo4yje2yqw4nDBEaNK2aDi4)
Sei
symmetrisch.
Symmetrische Tensoren haben reelle Eigenwerte und paarweise zueinander senkrechte oder orthogonalisierbare Eigenvektoren, die also eine Orthonormalbasis aufbauen. Die Eigenvektoren werden so nummeriert, dass sie ein Rechtssystem bilden.
Hauptachsentransformation mit Eigenwerten
und Eigenvektoren
des symmetrischen Tensors A:
![{\displaystyle {\begin{aligned}\mathbf {A} =&\sum _{i=1}^{3}\lambda _{i}{\hat {a}}_{i}\otimes {\hat {a}}_{i}=\left({\hat {a}}_{i}\otimes {\hat {e}}_{i}\right)\cdot \left(\sum _{j=1}^{3}\lambda _{j}{\hat {e}}_{j}\otimes {\hat {e}}_{j}\right)\cdot \left({\hat {e}}_{k}\otimes {\hat {a}}_{k}\right)\\=&{\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}\cdot {\begin{pmatrix}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&\lambda _{3}\end{pmatrix}}\cdot {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}^{\top }\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Co2zDajoQaqvFzts1ntJFoNwNaqhEajFAzNFAzNi4ntK2z2eNoqs5)
bzw.
![{\displaystyle {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}^{\top }\cdot \mathbf {A} \cdot {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}={\begin{pmatrix}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&\lambda _{3}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zqnFzNw1zNKPyqiNyja4zAs5ajzAnAvEoNeOatvEz2e5aNeOzNo0)
Sei
schiefsymmetrisch.
Schiefsymmetrische Tensoren haben einen reellen und zwei konjugiert komplexe, rein imaginäre Eigenwerte. Der reelle Eigenwert von A ist null zu dem ein Eigenvektor gehört, der proportional zur reellen #Vektorinvariante
ist. Siehe auch #Axialer Tensor oder Kreuzproduktmatrix.
Sei
und
eine Basis und
die dazu duale Basis.
Der Tensor
![{\displaystyle \mathbf {T} =a\,{\vec {a}}_{1}\otimes {\vec {a}}^{1}+b\,{\vec {a}}_{2}\otimes {\vec {a}}^{2}+c\,{\vec {a}}_{3}\otimes {\vec {a}}^{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ytFCaAe3nDw2z2hBzNzDnqsPajdDyqnFztCOatdBotvBots1zjhC)
hat die Eigenwerte
![{\displaystyle \lambda _{1}=a,\;\lambda _{2}=b,\;\lambda _{3}=c}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85njKOyqhAoDePoNvFoqwPags4aDa3aArBoqnFaDCOnqi4ntG4ygsN)
und Eigenvektoren
![{\displaystyle {\vec {v}}_{1}={\vec {a}}_{1},\;{\vec {v}}_{2}={\vec {a}}_{2},\;{\vec {v}}_{3}={\vec {a}}_{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9ByqdCzjmOnDvAz2i2z2vBytFFngrByqvAzta2nqi0zAzCoqdDyqdF)
Der #transponierte Tensor hat dieselben Eigenwerte zu den dualen Eigenvektoren
![{\displaystyle {\vec {v}}_{1}={\vec {a}}^{1},\;{\vec {v}}_{2}={\vec {a}}^{2},\;{\vec {v}}_{3}={\vec {a}}^{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OatvFaNdCztwPz2oNnjK0oAa0nAi4oteQa2ePoNzDoNoPoDvDaqa5)
Der Tensor
![{\displaystyle \mathbf {T} =c\,{\vec {a}}_{1}\otimes {\vec {a}}^{1}+a({\vec {a}}_{2}\otimes {\vec {a}}^{2}+{\vec {a}}_{3}\otimes {\vec {a}}^{3})+b({\vec {a}}_{2}\otimes {\vec {a}}^{3}-{\vec {a}}_{3}\otimes {\vec {a}}^{2})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BztdCaNBCntw5ate5ags4o2rByqs5oqzEagzEzDs5ztrFytK1ngrE)
hat die Eigenwerte
![{\displaystyle \lambda _{1}=c,\;\lambda _{2}=a+\mathrm {i} \,b,\;\lambda _{3}=a-\mathrm {i} \,b}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FnAi5ajm2aNG0zji4zNzDzAnFaqaOoNzDntBDnqeNztm1zAw0oDCN)
und Eigenvektoren
![{\displaystyle {\vec {v}}_{1}={\vec {a}}_{1},\;{\vec {v}}_{2}={\vec {a}}_{2}+\mathrm {i} \,{\vec {a}}_{3},\;{\vec {v}}_{3}={\vec {a}}_{2}-\mathrm {i} \,{\vec {a}}_{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Oo2wOoqdBzqdDzDo4ztrDzAo0agw4nthFztCOaNs5nto2oDe2zqnD)
Der #transponierte Tensor hat dieselben Eigenwerte zu den Eigenvektoren
![{\displaystyle {\vec {v}}_{1}={\vec {a}}^{1},\;{\vec {v}}_{2}={\vec {a}}^{2}-\mathrm {i} \,{\vec {a}}^{3},\;{\vec {v}}_{3}={\vec {a}}^{2}+\mathrm {i} \,{\vec {a}}^{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84yjnDztaPytiOoAnEnqzFnAdBajG5oNoNnghFnDG2oqo5nDG3ntwO)
Die #Eigenwerte
sind Invarianten.
Die Hauptinvarianten des Tensors A sind die Koeffizienten seines charakteristischen Polynoms:
![{\displaystyle \mathrm {det} (\mathbf {A} -x\mathbf {1} )=-x^{3}+\mathrm {Sp} (\mathbf {A} )x^{2}-\mathrm {I} _{2}(\mathbf {A} )x+\mathrm {det} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82oAi5zDK2yjmPnDC4zNdBoNi4aqs0nga0aDs2nAeOajlBnDm2zqvC)
Spezialfall:
![{\displaystyle \mathrm {det} ({\vec {b}}\otimes {\vec {c}}+a\mathbf {1} )=a^{2}(a+{\vec {b}}\cdot {\vec {c}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85oAe4oNe3nthDo2wNyto4oDo4aji5zgoPnAw3otG4zjC4ajeOzNBE)
Satz von Cayley-Hamilton:
![{\displaystyle -\mathbf {A} ^{3}+\mathrm {Sp} (\mathbf {A} )\mathbf {A} ^{2}-\mathrm {I} _{2}(\mathbf {A} )\mathbf {A} +\mathrm {det} (\mathbf {A} )\mathbf {1} =\mathbf {0} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82ajK2ygaQyqrAoDGOzAnDytFBatJAngw4aDzBzAe4ygdEzNnEyqe0)
Abbildung
![{\displaystyle \mathrm {Sp} (\mathbf {A} )=\mathrm {I} _{1}(\mathbf {A} )={\frac {1}{2}}(\mathbf {A} \#\mathbf {1} ):\mathbf {1} =\lambda _{1}+\lambda _{2}+\lambda _{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzDrDoAi1zqo5oDsQa2e2ztmOo2oNzArBz2o2oNe2othByqoQotoN)
mit #Eigenwerten λ1,2,3 von A.
![{\displaystyle \mathrm {Sp} ({\vec {a}}\otimes {\vec {g}})=\mathrm {Sp} ({\vec {g}}\otimes {\vec {a}}):={\vec {a}}\cdot {\vec {g}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80yjdEoNFDzDvEaDmNzDe4yjwQzjm2ngoNotw5aghAoDdCa2e3oDBB)
Linearität:
![{\displaystyle \mathrm {Sp} (\mathbf {A} )=\mathrm {Sp} (\mathbf {A} ^{\top })}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EoAi1oDo5oNe2yqa1nDnCajK5ngi1a2eOzti3ytC4ajaPoDiPajC3)
![{\displaystyle \mathrm {Sp} (\mathbf {A\cdot B} )=\mathrm {Sp} (\mathbf {B\cdot A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CytCQztFDzDo1z2vFzga1aqa1zNmOzNi3ajoNyjJCajzAathDaNC2)
![{\displaystyle \mathrm {Sp} (\mathbf {A^{\top }\cdot B} )=\mathrm {Sp} (\mathbf {A\cdot B^{\top }} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Angw0agaQaDGNzDFBzDe2aqa0ati0aqzBagsQzqoPztBAzjaNnDG4)
![{\displaystyle \mathrm {Sp} (\mathbf {A\cdot B\cdot C} )=\mathrm {Sp} (\mathbf {B\cdot C\cdot A} )=\mathrm {Sp} (\mathbf {C\cdot A\cdot B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85zjdEoDw2aqeNnDw2ztsOz2i4nAw5nAvBaqaOa2vCo2aNntrAnjK2)
In Komponenten:
![{\displaystyle \mathrm {Sp} \left(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)=A_{ii}=A_{11}+A_{22}+A_{33}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaqhDzqhBoNvBo2rDz2a2nji3yjaNatCNoDoOnjBEzts0zjvDnjm2)
![{\displaystyle \mathrm {Sp} \left(A^{ij}{\vec {a}}_{i}\otimes {\vec {b}}_{j}\right)=A^{ij}{\vec {a}}_{i}\cdot {\vec {b}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OoDG1zqw3aNiQotzEoNFBoAnFyqiOagi4oAoNoDmOaNo0zta4aqzE)
![{\displaystyle \mathrm {Sp} \left(A_{j}^{i}{\vec {a}}_{i}\otimes {\vec {a}}^{j}\right)=\mathrm {Sp} \left(A_{i}^{j}{\vec {a}}^{i}\otimes {\vec {a}}_{j}\right)=A_{i}^{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84a2zCnto0aghFzNG5zAzEatwOyjKNate3njdCoDJDztrDoto4z2wP)
Abbildung
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} ):={\frac {1}{2}}[\mathrm {Sp} (\mathbf {A} )^{2}-\mathrm {Sp} (\mathbf {A} ^{2})]={\frac {1}{2}}(\mathbf {A} \#\mathbf {A} ):\mathbf {1} =\lambda _{1}\lambda _{2}+\lambda _{2}\lambda _{3}+\lambda _{3}\lambda _{1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zgzFygrCygo1a2hAotdBoAw2aNmPoAwQoAhAyghCots4zjJDyqvE)
mit #Eigenwerten λ1,2,3 von A.
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} )=\mathrm {Sp(cof} (\mathbf {A} ))=\mathrm {Sp(adj} (\mathbf {A} ))}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qzqi4oAdFnqrDoAs3njG2zgePajJAatnCnDKQote1nDzCyteQatCO)
![{\displaystyle \mathrm {I} _{2}(x\mathbf {A} )=x^{2}\mathrm {I} _{2}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OaNe1yjzDyjJAyqvAnqo4ajoOngeNa2aQnDs4zNsQatsQz2zEoDa0)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} ^{\top })=\mathrm {I} _{2}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PzjFDoNvAnjBEygdCytGOajzDz2dDzNmPatG1nAsPoAiPzNJFyjoO)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} \cdot \mathbf {B} )=\mathrm {I} _{2}(\mathbf {B} \cdot \mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DztnEnqaOoNGQzNzEoDm2ztoPzAhEnjK5o2zFnje0aNvCote3oto1)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} \cdot \mathbf {B} \cdot \mathbf {C} )=\mathrm {I} _{2}(\mathbf {B} \cdot \mathbf {C} \cdot \mathbf {A} )=\mathrm {I} _{2}(\mathbf {C} \cdot \mathbf {A} \cdot \mathbf {B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BatdDatlFaqo2njBDaNJDnjeQoqo5otwNntdEzge4ata1zAi1nqeO)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} +\mathbf {B} )=\mathrm {I} _{2}(\mathbf {A} )+\mathrm {I} _{2}(\mathbf {B} )+\mathrm {Sp} (\mathbf {A} )\mathrm {Sp} (\mathbf {B} )-\mathrm {Sp} (\mathbf {A} \cdot \mathbf {B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OoqhDzNzAyjG3nDmOnDK1njC5yjJAoAvFnDnFajiOaAoQnjmPytaO)
In Komponenten:
![{\displaystyle \operatorname {I} _{2}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{12}A_{21}-A_{13}A_{31}-A_{23}A_{32}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CaNFBoDBFoteQngwQngw0oDwQz2s3ygzEzqsPnqnEyta3njs3oNdC)
![{\displaystyle \operatorname {I} _{2}(A^{ij}{\vec {a}}_{i}\otimes {\vec {b}}_{j})={\frac {1}{2}}A^{ij}A^{kl}[({\vec {a}}_{i}\cdot {\vec {b}}_{j})({\vec {a}}_{k}\cdot {\vec {b}}_{l})-({\vec {a}}_{i}\cdot {\vec {b}}_{l})({\vec {a}}_{k}\cdot {\vec {b}}_{j})]}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85oAsOaqoPytoNztw0njzBzNBEzgo3zgs2nAnEoNzEaqvDngzEzDzC)
![{\displaystyle \operatorname {I} _{2}\left(A_{j}^{i}{\vec {a}}_{i}\otimes {\vec {a}}^{j}\right)={\frac {1}{2}}(A_{i}^{i}A_{j}^{j}-A_{j}^{i}A_{i}^{j})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84aDoPa2s0o2vAnjdBoNG5aAa0aqzAztnFnjoPzNG2yqzBo2o1zDm5)
Abbildung
![{\displaystyle \mathrm {I} _{3}(\mathbf {A} ):=\mathrm {det} (\mathbf {A} )={\frac {1}{6}}(\mathbf {A} \#\mathbf {A} ):\mathbf {A} =\lambda _{1}\lambda _{2}\lambda _{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzDK5yqaOothFagzDzDC2ytm0yqa1oqrDnDG5oDCQzjmPoDFAoAoO)
mit #Eigenwerten λ1,2,3 von A.
![{\displaystyle \mathrm {det} (\mathbf {A} ^{\top })=\mathrm {det} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82aNJBygdEnDK5aDK3ntK2njnCoDe5nAhFntiNz2aOoDK5yqeOaqi1)
Determinantenproduktsatz:
![{\displaystyle \mathrm {det} (\mathbf {A\cdot B} )=\mathrm {det} (\mathbf {B\cdot A} )=\mathrm {det} (\mathbf {A} )\mathrm {det} (\mathbf {B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qz2oOzqzBagoQoDe2yjmOago4zNhCoAo3oqsOatKPzNeNatoQaqs2)
![{\displaystyle \mathrm {det} (\mathbf {A\cdot B\cdot C} )=\mathrm {det} (\mathbf {B\cdot C\cdot A} )=\mathrm {det} (\mathbf {C\cdot A\cdot B} )=\mathrm {det} (\mathbf {A} )\mathrm {det} (\mathbf {B} )\mathrm {det} (\mathbf {C} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QoDsNztGPa2e0z2aPoDFAoti1nqsNajdBzNoOaNK1atlFzDG4zjeP)
![{\displaystyle \mathrm {det} (\mathbf {A} ^{-1})={\frac {1}{\mathrm {det} (\mathbf {A} )}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BzgaPzts1yqw1zAo0o2aPoqw5zDi4zjwNnDo2nDvCaNGQnjo1nqzE)
Multiplikation mit Skalaren
:
![{\displaystyle {\begin{vmatrix}x{\vec {a}}&{\vec {b}}&{\vec {c}}\end{vmatrix}}={\begin{vmatrix}{\vec {a}}&x{\vec {b}}&{\vec {c}}\end{vmatrix}}={\begin{vmatrix}{\vec {a}}&{\vec {b}}&x{\vec {c}}\end{vmatrix}}=x{\begin{vmatrix}{\vec {a}}&{\vec {b}}&{\vec {c}}\end{vmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Bz2oQoDFEaDnBz2wNaAw3a2sNntJEoqs2ztzFaghFaAwQytnBa2vE)
![{\displaystyle \mathrm {det} (x\mathbf {A} )=x^{3}\mathrm {det} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85agw5oNiQa2eNzDCQnjiOnAaPz2oOaqi5yqoQaNm4ajBCaNKOoDi1)
In Komponenten:
![{\displaystyle {\begin{aligned}\mathrm {det} \left(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\right)=&{\begin{vmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{vmatrix}}\\=&A_{11}(A_{22}A_{33}-A_{23}A_{32})+A_{12}(A_{23}A_{31}-A_{21}A_{33})\\&+A_{13}(A_{21}A_{32}-A_{22}A_{31})\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QoDw4o2wQygeOo2zFzthAaAeQajnCnDFBoqrAaAvCa2iPzAs2zNG2)
![{\displaystyle {\begin{aligned}\mathrm {det} (A^{ij}{\vec {a}}_{i}\otimes {\vec {g}}_{j})=&{\begin{vmatrix}A^{11}&A^{12}&A^{13}\\A^{21}&A^{22}&A^{23}\\A^{31}&A^{32}&A^{33}\end{vmatrix}}{\begin{vmatrix}{\vec {a}}_{1}&{\vec {a}}_{2}&{\vec {a}}_{3}\end{vmatrix}}{\begin{vmatrix}{\vec {g}}_{1}&{\vec {g}}_{2}&{\vec {g}}_{3}\end{vmatrix}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85yta5zjaQoqs4aNw3zghCzDvBnDKPota4nte4zjo4oDo1z2ePygiN)
![{\displaystyle \operatorname {det} \left(A_{j}^{i}{\vec {a}}_{i}\otimes {\vec {a}}^{j}\right)={\begin{vmatrix}A_{1}^{1}&A_{2}^{1}&A_{3}^{1}\\A_{1}^{2}&A_{2}^{2}&A_{3}^{2}\\A_{1}^{3}&A_{2}^{3}&A_{3}^{3}\end{vmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Cnqo1ngiNzjKPoAnEzAvDntlDoAs1zDs1nDdCathEngnEnjs4ythF)
Zusammenhang mit den anderen Hauptinvarianten:
![{\displaystyle {\begin{aligned}\mathrm {det} (\mathbf {A} )=&{\frac {1}{6}}[\mathrm {Sp} (\mathbf {A} )^{3}-3\mathrm {Sp} (\mathbf {A} )\mathrm {Sp} (\mathbf {A} ^{2})+2\mathrm {Sp} (\mathbf {A} ^{3})]\\[1ex]=&{\frac {1}{3}}[\mathrm {Sp} (\mathbf {A} ^{3})+3\mathrm {Sp} (\mathbf {A} )\mathrm {I} _{2}(\mathbf {A} )-\mathrm {Sp} (\mathbf {A} )^{3}]\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AaDJFatlEoto4ngiPz2e2a2eOyjnFz2wOzNFCoqs0ajiPnDw2ats3)
Zusammenhang mit dem Spatprodukt:
![{\displaystyle (\mathbf {A} \cdot {\vec {a}})\cdot [(\mathbf {A} \cdot {\vec {b}})\times (\mathbf {A} \cdot {\vec {c}})]=\mathrm {det} (\mathbf {A} ){\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84aNGNaqw3zqwNzghEnDo1aDFDaqzAyqvDaNK1nDK1oDm0zjnCngoQ)
Zusammenhang mit #Äußeres Tensorprodukt:
![{\displaystyle \det(\mathbf {A} )={\frac {1}{6}}(\mathbf {A} \#\mathbf {A} ):\mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AnjJAoDsNyjBBaNmNzAhCaAiPaNdDnAa5aDBFnghFzqa0aDo1aDe4)
![{\displaystyle {\begin{aligned}\rightarrow \det(\mathbf {A+B} )=&\det(\mathbf {A} )+\det(\mathbf {B} )+\mathrm {Sp} (\mathbf {A} )\mathrm {I} _{2}(\mathbf {B} )+\mathrm {I} _{2}(\mathbf {A} )\mathrm {Sp} (\mathbf {B} )\\&+\mathrm {Sp} (\mathbf {A\cdot B\cdot (A+B)} )-\mathrm {Sp} (\mathbf {A\cdot B} )\mathrm {Sp} (\mathbf {A+B} )\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Bzje2oqrCztrEoqo3oqrAnta0ytBDatw1otm2aDnDyjhFoqs5aqw3)
Zusammenhang mit dem #Kofaktor:
![{\displaystyle \det(\mathbf {A} +\mathbf {B} )=\det(\mathbf {A} )+\mathrm {cof} (\mathbf {A} ):\mathbf {B} +\mathbf {A} :\mathrm {cof} (\mathbf {B} )+\det(\mathbf {B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82ntC2nAiQntmPngrCa2wNaAe0ytaQoNmQyti3aDlCz2o2otm0yqhB)
Abbildung
![{\displaystyle \parallel \mathbf {A} \parallel :={\sqrt {\mathbf {A} :\mathbf {A} }}={\sqrt {\mathrm {Sp} (\mathbf {A} ^{\top }\cdot \mathbf {A} )}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83nAdEa2zAztBAnqhCzgiQaDs1nDe2oDzFnqoNnja1oDa4nDhDate2)
![{\displaystyle \parallel {\vec {a}}\otimes {\vec {g}}\parallel =|{\vec {a}}|\,|{\vec {g}}|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BnAe1oNhCatK4ota4oNzAaqoNaAsPntCNz2s1ygi1aNo3ytsQoAaQ)
![{\displaystyle \parallel A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\parallel ={\sqrt {A_{ij}A_{ij}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OotzAngi5ztvAzjFCnAvAzAnAoqoOzte2zqnAaArDaDK2ztvDnjs2)
![{\displaystyle \parallel A^{ij}{\vec {a}}_{i}\otimes {\vec {g}}_{j}\parallel ={\sqrt {A^{ij}A^{kl}({\vec {a}}_{i}\cdot {\vec {a}}_{k})({\vec {g}}_{j}\cdot {\vec {g}}_{l})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eatw4zNC4yqvEnqsQzAi2oqa0oNKNzjCPnjK0njCPzjrBo2e5zNo1)
![{\displaystyle \parallel A_{j}^{i}{\vec {a}}_{i}\otimes {\vec {a}}^{j}\parallel ={\sqrt {A_{j}^{i}A_{l}^{k}({\vec {a}}_{i}\cdot {\vec {a}}_{k})({\vec {a}}^{j}\cdot {\vec {a}}^{l})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zDG5aNlDotJFoqhEaDdEoAoOotC0aArFztFBaAzByja4ngoQnjCQ)
Falls
:
![{\displaystyle \quad \parallel \mathbf {A} \parallel ={\sqrt {\mathrm {Sp} ^{2}(\mathbf {A} )-2\mathrm {I} _{2}(\mathbf {A} )}}={\sqrt {\mathrm {Sp} (\mathbf {A} ^{2})}}={\sqrt {\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnjnDyts5oqsQntFFntrAzqdDz2e4ajsNotzFzjnFa2aOyjK2oNw1)
Falls
:
![{\displaystyle \quad \parallel \mathbf {A} \parallel ={\sqrt {2\mathrm {I} _{2}(\mathbf {A} )}}={\sqrt {-\mathrm {Sp} (\mathbf {A} ^{2})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zNdCntiQotBFaDa1ngsPythFzqrEnjrCnjrDnte1aqo2yqo5aDdA)
Für #Schiefsymmetrische Tensoren
gibt es einen dualen axialen
Vektor
für den gilt:
für alle ![{\displaystyle {\vec {v}}\in \mathbb {V} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85zgdFntCPzDiNyqnAagaOaDnBaqo1nAa4otC4yjoQzqoNo2aNo2sN)
Der duale axiale Vektor ist proportional zur #Vektorinvariante:
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}:=-{\frac {1}{2}}{\vec {\mathrm {i} }}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnDeNaDKQaAhEytw4njrEygiPztsQnDe3otdEa2i1otGPyqnBoAe3)
Berechnung mit #Fundamentaltensor 3. Stufe
, #Kreuzprodukt von Tensoren oder #Skalarkreuzprodukt von Tensoren:
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}=-{\frac {1}{2}}{\stackrel {3}{\mathbf {E} }}:\mathbf {A} =-{\frac {1}{2}}\mathbf {A} \times \mathbf {1} =-{\frac {1}{2}}\mathbf {A} \cdot \!\!\times \mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DzNsNajlCzgw3ots3zDlEajaOnjrBajhFnDC2yjJEzjlCyghAngo5)
![{\displaystyle {\stackrel {A}{\overrightarrow {A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}}}}=-{\frac {1}{2}}A_{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}={\frac {1}{2}}{\begin{pmatrix}A_{32}-A_{23}\\A_{13}-A_{31}\\A_{21}-A_{12}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Oa2vAyti4nDhEatw2zgoPzgnEzDFFaNJBzqaNoqi1zqi1oNK3aja2)
![{\displaystyle {\stackrel {A}{\overrightarrow {A^{ij}({\vec {a}}_{i}\otimes {\vec {b}}_{j})}}}=-{\frac {1}{2}}A^{ij}{\vec {a}}_{i}\times {\vec {b}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84atCOzqw4zte5ntC0aAoNngdBoDJBzgaOajC2zqzFajGOoNdDotsO)
#Symmetrische Tensoren und #Kugeltensoren haben keinen dualen axialen Vektor:
Ein #Symmetrischer Anteil oder #Kugelanteil trägt nichts zum dualen axialen Vektor bei:
Seien x eine beliebige Zahl,
beliebige Vektoren und A, B beliebige Tensoren zweiter Stufe. Dann gilt:
![{\displaystyle {\stackrel {A}{\overrightarrow {{\vec {u}}\otimes {\vec {v}}}}}\;={\frac {1}{2}}{\vec {v}}\times {\vec {u}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82z2i3yjdFaNw5yts5a2i3zjaNzNa3oDsOntlDa2s5o2w3oAvAzDC1)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}\times {\vec {v}}=\mathbf {A} ^{\mathrm {A} }\cdot {\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnDK1zNe4oNhAygzAzDGPnqrCz2nCaDG1ztrEyge2zge3aAe0oNeO)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} ^{\top }}}}\quad \;=-{\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fz2nFoqsNnDwNato4zNzFnjJDygzEoArCyjJEaDBFyjoPztFDaDi1)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A+B} }}}={\stackrel {A}{\overrightarrow {\mathbf {A} }}}+{\stackrel {A}{\overrightarrow {\mathbf {B} }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaNmOzAs3zNiPzDzAzDoNygdDaDoPotm4zts0zNC2ags2zAzFyjlF)
![{\displaystyle {\stackrel {A}{\overrightarrow {x\mathbf {A} }}}\quad \;=x\,{\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DytdBzjnEzDmNzjvFnjw1aNG3aNvEoDKPyje2a2rBatePnghCoDrB)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} \#\mathbf {B} }}}\;=\mathbf {A} \cdot {\stackrel {A}{\overrightarrow {\mathbf {B} }}}+\mathbf {B} \cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80oNCOyjFCaqs5otmOngnCaDzCnDC3o2rDztzDzDs4oNnAzNmQo2dE)
![{\displaystyle \mathbf {A} \cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}\;=\mathbf {A} ^{\top }\cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}={\stackrel {A}{\overrightarrow {\mathbf {A} }}}\cdot \mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zDvEaNmQnDCNzts3zNsNoqaPztKPzDG3zge0yqnEztrEatCQnqrE)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathrm {cof} (\mathbf {A} )}}}=\mathbf {A} \cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84aDK1zje0ate3aNoNyqwQajm1oAs2atBCnqiQaNG2nDBAajeOajw3)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} ^{-1}}}}\quad =-{\frac {1}{\mathrm {det} (\mathbf {A} )}}\mathbf {A} \cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}\quad {\text{falls}}\quad \mathrm {det} (\mathbf {A} )\neq 0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnqdAnqe0otG5atzCnAo2z2vAoDBFaqw5nDaPygsOyghEa2zCoDdE)
![{\displaystyle {\stackrel {A}{\overrightarrow {{\vec {v}}\times \mathbf {A} }}}={\frac {1}{2}}(\mathrm {Sp} (\mathbf {A} )\mathbf {1} -\mathbf {A} )\cdot {\vec {v}}={\frac {1}{2}}(\mathbf {A} ^{\top }\#\mathbf {1} )\cdot {\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaNiNnjo3oAe5oDa1atePo2o5aqa3nqrCoNK0aqiQngi5a2iPotFE)
![{\displaystyle {\stackrel {A}{\overrightarrow {{\vec {v}}\times \mathbf {1} }}}\;\;={\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OoNCPaAi2yjdBzDzFotvEaqiNz2oQaAo2aAiOnDhDotdCzDa1z2hF)
![{\displaystyle {\stackrel {A}{\overrightarrow {({\vec {u}}\times {\vec {v}})\times \mathbf {A} }}}={\frac {1}{2}}({\vec {u}}\cdot \mathbf {A} \times {\vec {v}}-{\vec {v}}\cdot \mathbf {A} \times {\vec {u}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Coqi4oDdBo2aPoDhEnDFBaAe0zjGQzNBDyjFBa2e0yjrDyqdEajs1)
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {B\cdot A\cdot B} ^{\top }}}}\;\;=\mathrm {cof} (\mathbf {B} )\cdot {\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NagdAyjePajBCo2s5yjK5oDJAatsNajs5o2w0otC2zgs1nqwPa2zA)
Darin ist „#“ ein #Äußeres Tensorprodukt, cof(·) ist der #Kofaktor.
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} ):=\mathbf {A} \cdot \!\!\times \mathbf {1} =\mathbf {A} \times \mathbf {1} ={\stackrel {3}{\mathbf {E} }}:\mathbf {A} =-2{\stackrel {A}{\overrightarrow {\mathbf {A} }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaqwOyqsPoDnCyqaNaDeOoAi0zteNoqiNaDBEyjG1aDBFzjm5zgw1)
![{\displaystyle {\vec {\mathrm {i} }}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}={\begin{pmatrix}A_{23}-A_{32}\\A_{31}-A_{13}\\A_{12}-A_{21}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82oqdFygzEaAoOnDCQzDwNyqiNoDi0aDs4a2a1oNsOnDFFytvCzqi0)
![{\displaystyle {\vec {\mathrm {i} }}(A^{ij}({\vec {a}}_{i}\otimes {\vec {b}}_{j}))=A^{ij}{\vec {a}}_{i}\times {\vec {b}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80yjvAoDm3ntzFaNoNoDFDaqrDygo2ajKQzqoPzNKQzNwOzjG4yjs2)
Zusammenhang mit dem #Skalarkreuzprodukt von Tensoren:
![{\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} ={\vec {\mathrm {i} }}(\mathbf {A} \cdot \mathbf {B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaqvFytmQnqi3oNoNzjzFzqvAntG5oDBEaghEaqsNnDm3zqnEzAaN)
#Symmetrische Tensoren haben keine Vektorinvariante:
Die Eigenschaften des dualen axialen Vektors sind hierher übertragbar. Seien x eine beliebige Zahl,
beliebige Vektoren und A, B beliebige Tensoren zweiter Stufe. Dann gilt:
![{\displaystyle {\vec {\mathrm {i} }}({\vec {u}}\otimes {\vec {v}})\;\;={\vec {u}}\times {\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OajBEnDK0ate1zgdEnjm0oDeNotvEnqhAnje1oNsPa2w2zjG2zAhA)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} )\times {\vec {v}}\;=-2\mathbf {A} ^{\mathrm {A} }\cdot {\vec {v}}=(\mathbf {A^{\top }-A} )\cdot {\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QyqsNnqaOztlFzDmOnjaNajnEaDo4aDBFngsNyjwOajaPaNhEaNe4)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} ^{\top })\quad \;=-{\vec {\mathrm {i} }}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Doqe3yja3zto4oqi3oNs1ajBEaNm5oAzEaNiQathFoqaQoNzAaAnD)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A+B} )={\vec {\mathrm {i} }}(\mathbf {A} )+{\vec {\mathrm {i} }}(\mathbf {B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AygnDnteOoAaOzjiOnjvDagrEyje3zje4ngw3oqw2nDw5aAa2oqaN)
![{\displaystyle {\vec {\mathrm {i} }}(x\mathbf {A} )\quad \;=x\,{\vec {\mathrm {i} }}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85oqvAzgrDaNKOnDo1oNe0oDJDzDs4zqnCz2s4a2vEnqrDygs3ztBC)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} \#\mathbf {B} )\;=\mathbf {A} \cdot {\vec {\mathrm {i} }}(\mathbf {B} )+\mathbf {B} \cdot {\vec {\mathrm {i} }}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fnqa0zNwNatoQo2zBaNa3yqi4aNwQnqiNygrBzts3ajJEnAo4oNe0)
![{\displaystyle \mathbf {A} \cdot {\vec {\mathrm {i} }}(\mathbf {A} )\;=\mathbf {A} ^{\top }\cdot {\vec {\mathrm {i} }}(\mathbf {A} )={\vec {\mathrm {i} }}(\mathbf {A} )\cdot \mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzDK3zgaQagnEzqzBoDdByga2zNi0z2hCyjdDnAi2njaPajGOoAw0)
![{\displaystyle {\vec {\mathrm {i} }}(\mathrm {cof} (\mathbf {A} ))=\mathbf {A} \cdot {\vec {\mathrm {i} }}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DoDsQagzFzqdAnjrBzqs3otdAyjG2zgs2aDe3yjvCotoNajo0ygi3)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} ^{-1})\quad =-{\frac {1}{\mathrm {det} (\mathbf {A} )}}\mathbf {A} \cdot {\vec {\mathrm {i} }}(\mathbf {A} )\quad {\text{falls}}\quad \mathrm {det} (\mathbf {A} )\neq 0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82o2eQyjeNoNrFntm2zjKQoAaQoDCNyqhDntsNaAzBajePoDe2njlA)
![{\displaystyle {\vec {\mathrm {i} }}({\vec {v}}\times \mathbf {A} )\;=(\mathbf {A} -\mathrm {Sp} (\mathbf {A} )\mathbf {1} )\cdot {\vec {v}}=-(\mathbf {A} ^{\top }\#\mathbf {1} )\cdot {\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PaAi3zqo3zDBDotrAzDJFzAa2nDrDzji0zts2atiPzNFFnqnBajdD)
![{\displaystyle {\vec {\mathrm {i} }}({\vec {v}}\times \mathbf {1} )\;\;=-2{\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zDePytrCoAdFzqdAnjvFoAhCaDePzghEaAi3nqw0ntzEaqdDzjnC)
![{\displaystyle {\vec {\mathrm {i} }}(({\vec {u}}\times {\vec {v}})\times \mathbf {A} )={\vec {v}}\cdot \mathbf {A} \times {\vec {u}}-{\vec {u}}\cdot \mathbf {A} \times {\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aDK0njw4nDlEzNCQoDK1oDmPyjo4atBAo2hDnjzAz2w2yqdCntC4)
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {B\cdot A\cdot B} ^{\top })=\mathrm {cof} (\mathbf {B} )\cdot {\vec {\mathrm {i} }}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaNG3zAnFa2s1nqvAoqs0otKOatnBaNmQzDC5z2wNyjnBaDwPygoP)
Darin ist „#“ ein #Äußeres Tensorprodukt, cof(·) ist der #Kofaktor.
Definition
![{\displaystyle \mathbf {A} :={\vec {a}}\otimes {\vec {b}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dz2sPnjzCzNoQzAaQatwNzjeQzAwQzjzDoAo1ntvCzNo3oDvDo2dE)
Kofaktor:
#Invarianten:
![{\displaystyle \mathrm {Sp} (\mathbf {A} )={\vec {a}}\cdot {\vec {b}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zNe3otnCzgvCytaQnDi1aDiNaNiPnjsQaqdAoDwOnjnCyqaPnti5)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} )=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ntJFoNwNaAhCoDwQzDeNzqzCzjdCajKQzAo3nqw4ytdFagdAoqvE)
![{\displaystyle \mathrm {det} (\mathbf {A} )=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Doqo3nDC5aDC4ztlAz2w1njzFyjCOzDJBnAe4ythDyqeOyjaOaDa1)
![{\displaystyle \parallel \mathbf {A} \parallel =|{\vec {a}}|\,|{\vec {b}}|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AytnCnDiPzDhFzDG0ytzBntdEyqdBzji3oAnAoNo3ztw0aNw1zDvA)
#Eigensystem:
![{\displaystyle {\begin{aligned}\lambda _{1}=&{\vec {a}}\cdot {\vec {b}},&{\vec {v}}_{1}=&{\frac {\vec {a}}{|{\vec {a}}|}}\\\lambda _{2}=&0,&{\vec {v}}_{2}=&{\frac {{\vec {a}}\times {\vec {b}}}{|{\vec {a}}\times {\vec {b}}|}}\\\lambda _{3}=&0,&{\vec {v}}_{3}=&{\frac {({\vec {a}}\times {\vec {b}})\times {\vec {b}}}{|({\vec {a}}\times {\vec {b}})\times {\vec {b}}|}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BzDw2ztaNzjo0aNrBnDdAaAw0zgeNnjCOaDK5nDK3aga0nta2z2nD)
Gegeben ein beliebiger Tensor 2. Stufe A. Dieser kann immer als Summe dreier Dyaden dargestellt werden:
![{\displaystyle {\begin{aligned}\mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}=&{\vec {s}}_{j}\otimes {\hat {e}}_{j}={\begin{pmatrix}{\vec {s}}_{1}&{\vec {s}}_{2}&{\vec {s}}_{3}\end{pmatrix}}\\=&{\hat {e}}_{i}\otimes {\vec {z}}_{i}={\begin{pmatrix}{\vec {z}}_{1}&{\vec {z}}_{2}&{\vec {z}}_{3}\end{pmatrix}}^{\top }\\=&{\vec {a}}_{k}\otimes {\vec {g}}_{k}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AytBDz2nEyja5otCOotaQzDi2aNG5o2a1otJCoDaPzDo4zNnCoDo2)
mit Spaltenvektoren
, Zeilenvektoren
und
.
#Hauptinvarianten (
):
![{\displaystyle \mathrm {I} _{1}(\mathbf {A} )=s_{i,i}=z_{i,i}={\vec {a}}_{i}\cdot {\vec {g}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83oAi2aqnDytBDoNmNzNlAzDhFajrDzgoQnDrFzNo3zNnFyqzEz2a1)
![{\displaystyle {\begin{aligned}\mathrm {I} _{2}(\mathbf {A} )=&{\frac {1}{2}}(s_{i,i}s_{j,j}-s_{i,j}s_{j,i})={\frac {1}{2}}(z_{i,i}z_{j,j}-z_{i,j}z_{j,i})\\=&{\frac {1}{2}}[({\vec {a}}_{i}\cdot {\vec {g}}_{i})({\vec {a}}_{j}\cdot {\vec {g}}_{j})-({\vec {a}}_{i}\cdot {\vec {g}}_{j})({\vec {a}}_{j}\cdot {\vec {g}}_{i})]\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83agdAyje0ngnAajmPoqhBoAsNzDJCygdDoNeQztmOajG4oAzDytsN)
![{\displaystyle \mathrm {I} _{3}(\mathbf {A} )={\begin{vmatrix}{\vec {s}}_{1}&{\vec {s}}_{2}&{\vec {s}}_{3}\end{vmatrix}}={\begin{vmatrix}{\vec {z}}_{1}&{\vec {z}}_{2}&{\vec {z}}_{3}\end{vmatrix}}={\begin{vmatrix}{\vec {a}}_{1}&{\vec {a}}_{2}&{\vec {a}}_{3}\end{vmatrix}}{\begin{vmatrix}{\vec {g}}_{1}&{\vec {g}}_{2}&{\vec {g}}_{3}\end{vmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzqzAntC1zgs1aqw2oDw5aNGPytoOoqw0aqs5ztG0zjeQatm1aNdC)
#Betrag:
![{\displaystyle \parallel \mathbf {A} \parallel ={\sqrt {{\vec {s}}_{i}\cdot {\vec {s}}_{i}}}={\sqrt {{\vec {z}}_{i}\cdot {\vec {z}}_{i}}}={\sqrt {({\vec {a}}_{i}\cdot {\vec {a}}_{j})({\vec {g}}_{i}\cdot {\vec {g}}_{j})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QatdEaDCNygvDoAw5zAdAzAe4zNi1oNG2aNi2aDFFzqnCzNwPztnF)
#Dualer axialer Vektor:
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} }}}={\frac {1}{2}}{\hat {e}}_{i}\times {\vec {s}}_{i}={\frac {1}{2}}{\vec {z}}_{i}\times {\hat {e}}_{i}={\frac {1}{2}}{\vec {g}}_{i}\times {\vec {a}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NnjdEaAnCaNa5aAsNoqsNaDKQotC5zDoNagsNzts0yqe5atsQnjo4)
#Vektorinvariante:
![{\displaystyle {\vec {\mathrm {i} }}(\mathbf {A} )={\vec {s}}_{i}\times {\hat {e}}_{i}={\hat {e}}_{i}\times {\vec {z}}_{i}={\vec {a}}_{i}\times {\vec {g}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dntm0a2zFyjiNa2dBotiPzDe3ygwOytGOotoQnDlDatvEoNo2zAdD)
#Kofaktor:
![{\displaystyle {\begin{aligned}\mathrm {cof} (\mathbf {A} )=&{\vec {z}}_{i}\otimes {\vec {s}}_{i}-\mathrm {I} _{1}(\mathbf {A} )\mathbf {A} ^{\top }+\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} \\=&({\vec {a}}_{i}\cdot {\vec {g}}_{j}){\vec {g}}_{i}\otimes {\vec {a}}_{j}-({\vec {a}}_{i}\cdot {\vec {g}}_{i}){\vec {g}}_{j}\otimes {\vec {a}}_{j}+\mathrm {I} _{2}(\mathbf {A} )\mathbf {1} \end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PaNe3a2e4oAa0z2iOzgw4ntsOz2vDnAe4zqhEz2i1yja0yjsOo2nF)
#Inverse:
![{\displaystyle \mathbf {A} ^{-1}={\hat {e}}_{i}\otimes {\vec {s}}^{i}={\vec {z}}^{i}\otimes {\hat {e}}_{i}={\vec {g}}^{i}\otimes {\vec {a}}^{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Dyta1yqzBaArDatoQo2i5otdBzto4aDhEyjJBo2eNo2dBaNi3yje3)
![{\displaystyle \mathbf {1} ={\hat {e}}_{i}\otimes {\hat {e}}_{i}=\delta _{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzDo2yjs4ztBEnjmOaDoOzqdCzDmNytmPajdFoDvEoNGOytBAyjzF)
![{\displaystyle \mathbf {1} ={\vec {g}}_{i}\otimes {\vec {g}}^{i}={\vec {g}}^{i}\otimes {\vec {g}}_{i}=g^{ij}{\vec {g}}_{i}\otimes {\vec {g}}_{j}=g_{ij}{\vec {g}}^{i}\otimes {\vec {g}}^{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaghCaNJEaAw2nge0zqnCnge2aAs4zgi2njdBz2rFaDvEoNi4ato4)
mit
Allgemein:
![{\displaystyle \mathbf {1} =({\vec {a}}^{i}\cdot {\vec {g}}^{j}){\vec {a}}_{i}\otimes {\vec {g}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QytG5nqnDzNhAyge4zDiPzqa2zDo1nAw1nDm1yqhEntK5z2wQnDzD)
#Transposition und #Inverse:
![{\displaystyle \mathbf {1} =\mathbf {1} ^{\top }=\mathbf {1} ^{-1}=\mathbf {1} ^{\top -1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81oDsNnti1otC4oDlEzAs3aNrEats3njK0a2i4otsPz2hAajeQyqaQ)
Kofaktor:
Vektortransformation
![{\displaystyle \mathbf {1} \cdot {\vec {v}}={\vec {v}}\cdot \mathbf {1} ={\vec {v}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85oAnFnjw3nqaOatmNoDFAzti1oqrCntGQaghDzDa0ygeNztGPoDs4)
Tensorprodukt
![{\displaystyle \mathbf {A\cdot 1} =\mathbf {1\cdot A} =\mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ba2aQzAhCaNa4aAoQatJDnDJBoAzAaAzDnjzBzjC5oto3ntw3zja4)
Skalarprodukt
![{\displaystyle \mathbf {A} :\mathbf {1} =\mathrm {Sp} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83oAe4yqw0zje4yqnAzti0zqwQnDJAaDa4zNhAoAwPnqw2zqnEajsP)
#Invarianten:
![{\displaystyle \mathrm {Sp} (\mathbf {1} )=\mathbf {1} :\mathbf {1} =3}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eygs3aNi5oAnFaqzFzDC3aNBAoNG5ztzFnjFDaNK3agwPoDG5oNFE)
![{\displaystyle \mathrm {I} _{2}(\mathbf {1} )=3}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zge1oNK1aqvCo2aOytvDnjBBzqe0aNmNzDFAzghFaDBAaNaNyjvA)
![{\displaystyle \mathrm {det} (\mathbf {1} )=1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Azgs2agsNnDw2ytdBaNe4o2s4atw1zNdBntCQnDhDytaNoDnFyqs1)
![{\displaystyle \parallel \mathbf {1} \parallel ={\sqrt {3}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Az2e5oNoPoNrFzNo3yqoOnjo5oqiOzgo1aqzEotzAaNJBzDzCzjwN)
#Eigenwerte:
![{\displaystyle \lambda _{1,2,3}=1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NztBCztK0yji0oti4yqhDagrDaja0zNw5atiPnjFEoDnEnDmPoDa3)
Alle Vektoren sind #Eigenvektoren.
Definition
![{\displaystyle \mathbf {H} :\quad \mathrm {det} (\mathbf {H} )=1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FaDFAzqnFzNJBnqzCa2w3ngs2yjK5zNlFagrAzqhDats3ataPyjC3)
Kofaktor:
Determinantenproduktsatz:
![{\displaystyle \mathrm {det} (\mathbf {A\cdot H} )=\mathrm {det} (\mathbf {H\cdot A} )=\mathrm {det} (\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83o2nFatJEagnBnthAzAwOats5atzDo2o4zAs5oqoNoDCOoNm0atnD)
Definition
![{\displaystyle \mathbf {Q} :\quad \mathbf {Q} ^{-1}=\mathbf {Q} ^{\top }\quad {\textsf {oder}}\quad \mathbf {Q\cdot Q} ^{\top }=\mathbf {Q} ^{\top }\cdot \mathbf {Q} =\mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AztBAa2sPyje0ztzCyti5oDe3otGQz2s1a2iPo2sPaArEntC5njrC)
Kofaktor:
#Invarianten (
ist der Drehwinkel):
![{\displaystyle \mathrm {Sp} (\mathbf {Q} )=\mathrm {det} (\mathbf {Q} )+2\cos(\alpha )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzNa5oqaQntw4aDhFzjo1zqeQntzFaghCz2a1njrFzqzFotlCnqoP)
![{\displaystyle \mathrm {I} _{2}(\mathbf {Q} )=\mathrm {det} (\mathbf {Q} )\cdot \mathrm {Sp} (\mathbf {Q} )=1+2\,\mathrm {det} (\mathbf {Q} )\cos(\alpha )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Po2vDaNJBnAe0zNa3oDsOotzAoNCQzDrDzAe1atm3yqa1ote1oDrD)
![{\displaystyle \mathrm {det} (\mathbf {Q} )=\pm 1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PaAnDatBCzAzCzje3ajCNythCztJEnqwPoAnEoNFBaNnCzDFEz2i0)
![{\displaystyle \parallel \mathbf {Q} \parallel ={\sqrt {3}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnDC4zNCQzArFaDJAyjC0aNmOzqe4zNlFygsOzDm0oAdEnjG2zNFB)
Eigentlich orthogonaler Tensor
, entspricht einer Drehung.
Uneigentlich orthogonaler Tensor
, entspricht einer Drehspiegelung.
Spatprodukt:
![{\displaystyle (\mathbf {Q} \cdot {\vec {a}})\cdot [(\mathbf {Q} \cdot {\vec {b}})\times (\mathbf {Q} \cdot {\vec {c}})]=\mathrm {det} (\mathbf {Q} ){\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zDKQoDK2zNi2ote0aqa1oNw3nthEzqzDoNa3nqeQote5yte1nje2)
Kreuzprodukt und #Kofaktor:
![{\displaystyle (\mathbf {Q} \cdot {\vec {a}})\times (\mathbf {Q} \cdot {\vec {b}})=\mathrm {det} (\mathbf {Q} )\mathbf {Q} \cdot ({\vec {a}}\times {\vec {b}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85yqrFnthEntFFzjsNoDlFyge2zNaNaDvDyteQz2e5otrEzje4aNiQ)
![{\displaystyle \mathrm {cof} (\mathbf {Q} )=\mathrm {det} (\mathbf {Q} )\mathbf {Q} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BnAs2zjaPygnAzqw4a2iQaNlAaDvDoNzDa2eNz2o2oqdBzjCOote3)
Gegeben ein Einheitsvektor
und Drehwinkel α. Dann sind die folgenden Tensoren R zueinander gleich, orthogonal und drehen um die Achse
mit Winkel α:
Rodrigues-Formel:
![{\displaystyle {\begin{aligned}\mathbf {R} =&\mathbf {1} +s_{\alpha }{\hat {n}}\times \mathbf {1} +d_{\alpha }({\hat {n}}\times \mathbf {1} )^{2}\\=&\mathbf {1} +s_{\alpha }{\hat {n}}\times \mathbf {1} +d_{\alpha }({\hat {n}}\otimes {\hat {n}}-\mathbf {1} )\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ba2hEajeNnjvDnqnCnAi0zAdDyqnDngwPaDiPytlEotrCnje4aDdF)
![{\displaystyle \mathbf {R} ={\begin{pmatrix}c_{\alpha }+d_{\alpha }n_{1}^{2}&-s_{\alpha }n_{3}+d_{\alpha }n_{1}n_{2}&s_{\alpha }n_{2}+d_{\alpha }n_{1}n_{3}\\s_{\alpha }n_{3}+d_{\alpha }n_{1}n_{2}&c_{\alpha }+d_{\alpha }n_{2}^{2}&-s_{\alpha }n_{1}+d_{\alpha }n_{2}n_{3}\\-s_{\alpha }n_{2}+d_{\alpha }n_{1}n_{3}&s_{\alpha }n_{1}+d_{\alpha }n_{2}n_{3}&c_{\alpha }+d_{\alpha }n_{3}^{2}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84oqaPz2sQoNGNnqs3oAoOyjFBoDi3oqs5ytJBzArBygoOoAi2zAiQ)
mit
.
Euler-Rodrigues-Formel:
also
:
![{\displaystyle \mathbf {R} :={\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(bc-ad)&2(bd+ac)\\2(bc+ad)&a^{2}+c^{2}-b^{2}-d^{2}&2(cd-ab)\\2(bd-ac)&2(cd+ab)&a^{2}+d^{2}-b^{2}-c^{2}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OoDrFaAw4ajaOzjwOzjJFagrEa2e3atzFzNa2o2iNotdAnAs3z2vD)
Formulierung mit Drehvektor:
Drehvektor |
|
Orthogonaler Tensor
|
![{\displaystyle {\vec {\alpha }}=\alpha {\vec {n}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81nga2atnAaAoPaNe2zAzCzDFDaAvFatCQzqs4njBFzNdAothDnDnC) |
→
|
|
![{\displaystyle {\vec {\alpha }}=\tan(\alpha ){\vec {n}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaDi1nAvFoNFByjvBaqzFztw5aqa2oAiOajm3ajCPnjFCyqw3atJF) |
→
|
|
|
→
|
|
![{\displaystyle {\vec {\alpha }}=\sin(\alpha )\;{\vec {n}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qoqe0oDJBnqvBaNo2aNG3zNdBzjhFytm1zjs0ataNzNJByta0oNC4) |
→
|
|
![{\displaystyle {\vec {\alpha }}=\sin \left({\frac {\alpha }{2}}\right)\;{\vec {n}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzDJBaAw4aghCoqvCo2i2zqw3ntC4nDFBythBoDm2zjC2zAhFyjs3) |
→
|
|
![{\displaystyle {\vec {\alpha }}=\cos(\alpha )\;{\vec {n}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Po2w3nAe0yje3ntG0oDnBz2e4nDaQntrAoDeNyjzFzNKOoNo3ageP) |
→
|
|
|
→
|
|
Darin ist
Beispiel für Drehspiegelung:
![{\displaystyle \mathbf {Q} =-\mathbf {1} +\sin(\alpha ){\hat {n}}\times \mathbf {1} -(1+\cos(\alpha ))({\hat {n}}\times \mathbf {1} )^{2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aArCajlCoDw0zqw2oDFDotBAzqwPygdDnAvEa2a2nDFFoAnFaNs4)
Drehung von Vektorraumbasis
mit Drehachse
:
![{\displaystyle \mathbf {Q} \cdot {\vec {u}}_{i}={\vec {v}}_{i}\,,\quad \mathbf {Q} \cdot {\vec {u}}^{i}={\vec {v}}^{i}\,,\quad \mathbf {Q} ={\vec {v}}_{i}\otimes {\vec {u}}^{i}={\vec {v}}^{i}\otimes {\vec {u}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CnjlEzjaNzNBCntaPaAw3oNKNaNvBaqhBata0ygi2o2vCzgi5ataO)
![{\displaystyle {\hat {n}}\simeq {\vec {v}}_{i}\times {\vec {u}}^{i}={\vec {v}}^{i}\times {\vec {u}}_{i}=-2{\stackrel {A}{\overrightarrow {\mathbf {Q} }}}={\vec {\mathrm {i} }}(\mathbf {Q} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85zAePzDdFaDlDzjwQzNFCzNBFzNm3nDCPytmNygs3yjvDnto2aghA)
mit #Dualer axialer Vektor
und #Vektorinvariante
.
Gegeben Orthonormalbasis
, Drehwinkel
und
ist Drehachse:
![{\displaystyle {\begin{aligned}\mathbf {Q} =&{\color {red}\pm }{\hat {v}}_{1}\otimes {\hat {v}}_{1}+\cos(\alpha )({\hat {v}}_{2}\otimes {\hat {v}}_{2}+{\hat {v}}_{3}\otimes {\hat {v}}_{3})+\sin(\alpha )({\hat {v}}_{3}\otimes {\hat {v}}_{2}-{\hat {v}}_{2}\otimes {\hat {v}}_{3})\\=&{\begin{pmatrix}{\color {red}\pm 1}&0&0\\0&\cos(\alpha )&-\sin(\alpha )\\0&\sin(\alpha )&\cos(\alpha )\end{pmatrix}}_{{\hat {v}}_{i}\otimes {\hat {v}}_{j}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FnAaQatK1yjGPnqdBzqaPoNeOa2dCaDw0othEyje3ataOaNKNoNzE)
: Drehung,
: Drehspiegelung um ![{\displaystyle {\hat {v}}_{1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81nDeQzNi0aNlAzNoPzNsQzga4njdDaAnAajo5ytCQnto1oAhCz2nC)
Wenn
ein Rechtssystem (Mathematik) bilden, dann dreht Q gegen den Uhrzeigersinn, sonst im Uhrzeigersinn um die Drehachse.
#Eigensystem:
![{\displaystyle {\begin{aligned}\lambda _{1}=&\mathrm {det} (\mathbf {Q} )\,,&{\vec {q}}_{1}=&{\hat {v}}_{1}\\\lambda _{2}=&e^{\mathrm {i} \alpha },&{\vec {q}}_{2}=&{\frac {1}{\sqrt {2}}}({\hat {v}}_{2}-\mathrm {i} {\hat {v}}_{3}).\\\lambda _{3}=&e^{-\mathrm {i} \alpha },&{\vec {q}}_{3}=&{\frac {1}{\sqrt {2}}}({\hat {v}}_{2}+\mathrm {i} {\hat {v}}_{3})\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AnAdEz2w2oNJDnqhAoDsQote0o2vByjm1oDi5ags4a2i2aNm0a2hD)
Drehwinkel:
![{\displaystyle \cos(\alpha )={\frac {1}{2}}(\mathrm {Sp} (\mathbf {Q} )-\mathrm {det} (\mathbf {Q} ))}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zgzFaDK2ats3yjo4ygiNajhFnjs3oArBaDK1aDiNaNs0yjsOnjaN)
Drehachse
ist #Vektorinvariante:
![{\displaystyle {\hat {n}}\simeq {\vec {\mathrm {i} }}(\mathbf {Q} )=\mathbf {1} \cdot \!\!\times \mathbf {Q} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Engi1yqaNntrDzDGNzNKPngsNnjaOzjrDzAnCaNwQoDrEzNmPaDi5)
![{\displaystyle \mathbf {Q} ={\vec {s}}_{i}\otimes {\vec {e}}_{i}={\vec {e}}_{i}\otimes {\vec {z}}_{i}\quad \rightarrow \quad {\hat {n}}\simeq {\vec {s}}_{i}\times {\vec {e}}_{i}={\vec {e}}_{i}\times {\vec {z}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CnDlCoDGNatlFoAwPaNFFyqa0ats5aDK2aga4aDsPnqa3oAsQageO)
![{\displaystyle {\frac {1}{2}}(\mathbf {Q} -\mathbf {Q} ^{\top })=\sin(\alpha ){\hat {n}}\times \mathbf {1} =\sin(\alpha ){\begin{pmatrix}0&-n_{3}&n_{2}\\n_{3}&0&-n_{1}\\-n_{2}&n_{1}&0\end{pmatrix}},\quad |{\hat {n}}|=1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83oDvAnjdAa2w5zDo1z2nFztlFoqoNztnFaAwQoNK5agePoNdFzthE)
Definition
![{\displaystyle \mathbf {A} :\quad {\vec {v}}\cdot \mathbf {A} \cdot {\vec {v}}>0\quad \forall \;{\vec {v}}\in \mathbb {V} \setminus \{{\vec {0}}\}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FaNKQagw1ngdEoDo1yga0z2w5ztC2yqwQzDhFztnByteNagi5otm1)
Kofaktor:
Notwendige Bedingungen für positive Definitheit:
![{\displaystyle \mathrm {det} (\mathbf {A} )>0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82a2s2oDrAo2aOaNFFoNm0o2zFotw0agvCyqa5ztK5ote4ngnFagvF)
![{\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\quad \rightarrow \quad A_{11},\,A_{22},\,A_{33}>0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ba2nBzAw5zNoPytdEnjs1yga1nqsOzjK0ytsPztlDo2wQntwNajw3)
![{\displaystyle \mathbf {A} =A_{j}^{i}{\vec {a}}_{i}\otimes {\vec {a}}^{j}\quad \rightarrow \quad A_{1}^{1},\,A_{2}^{2},\,A_{3}^{3}>0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PzDvFoDBFytePyqs3atdEzgrFygvFzjKOnjC1nDG2zDG0aqoOntCO)
Notwendige und hinreichende Bedingung für positive Definitheit: Alle #Eigenwerte von A sind größer als null.
Immer positiv definit falls det(A) ≠ 0:
- A·A⊤ und A⊤·A
Definition
![{\displaystyle \mathbf {A} :\quad \mathbf {A} =\mathbf {A} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DzDCOa2rEytdFagrFoNdEzDeNztoNnDlFzjJDoDJBatJEaDJEajFD)
Kofaktor:
#Betrag:
![{\displaystyle \quad \parallel \mathbf {A} \parallel ={\sqrt {\mathrm {Sp} ^{2}(\mathbf {A} )-2\mathrm {I} _{2}(\mathbf {A} )}}={\sqrt {\mathrm {Sp} (\mathbf {A} ^{2})}}={\sqrt {\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnjnDyts5oqsQntFFntrAzqdDz2e4ajsNotzFzjnFa2aOyjK2oNw1)
Bei Symmetrischen Tensoren verschwinden ihr #Dualer axialer Vektor und ihre #Vektorinvariante:
![{\displaystyle {\stackrel {A}{\overrightarrow {\mathbf {A} ^{\mathrm {S} }}}}={\vec {\mathrm {i} }}(\mathbf {A} ^{\mathrm {S} })={\vec {0}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82ajdEyjoNajFFoDi2oDBDoNKOnDa2zAdBzqsPzDGOaNaNzjKOntsO)
Bilinearform:
![{\displaystyle {\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}={\vec {v}}\cdot \mathbf {A} \cdot {\vec {u}}\quad \forall {\vec {u}},{\vec {v}}\in \mathbb {V} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EngzEaqw4oDFBajlEyjG0ajC2njsQoNBAnjlEzNCQajBDajhCzqdB)
Alle #Eigenwerte λ1,2,3 sind reell. Alle #Eigenvektoren
sind reell und paarweise orthogonal zueinander oder orthogonalisierbar. Hauptachsentransformation:
![{\displaystyle {\begin{aligned}\mathbf {A} =&\sum _{i=1}^{3}\lambda _{i}{\hat {a}}_{i}\otimes {\hat {a}}_{i}=({\hat {a}}_{i}\otimes {\hat {e}}_{i})\left(\sum _{i=1}^{3}\lambda _{j}{\hat {e}}_{j}\otimes {\hat {e}}_{j}\right)({\hat {e}}_{k}\otimes {\hat {a}}_{k})\\=&{\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}\cdot {\begin{pmatrix}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&\lambda _{3}\end{pmatrix}}\cdot {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}^{\top }\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fo2zAnta0nqvBoqs0ztsQytBCnqzBzNi1nqzEyqiPoti5ytdBoDGP)
Bezüglich der Standardbasis:
![{\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{12}&A_{22}&A_{23}\\A_{13}&A_{23}&A_{33}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PaAw0atw0nDGNnqnCnjK5ata0aDoPnqnDztw3oDzAztlDyqiNyta5)
#Invarianten:
![{\displaystyle \mathrm {Sp} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{11}+A_{22}+A_{33}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzDFBaDhBngw3yjJDoNwQaDnDoAaPaDhAoqsNajw5ytw2zge4zjKN)
![{\displaystyle \mathrm {I} _{2}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{12}^{2}-A_{13}^{2}-A_{23}^{2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zNKPntJAz2w5zqoQyqdDa2iOaAzAatFAaNw2nAw5zqdCntC2nga0)
![{\displaystyle {\begin{aligned}\mathrm {det} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=&A_{11}(A_{22}A_{33}-A_{23}^{2})+A_{12}(A_{23}A_{13}-A_{12}A_{33})\\&+A_{13}(A_{12}A_{23}-A_{13}A_{22})\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84atdFzNaNa2e4ngi1ajG2zAzFnjaPagoOoqs4aAoPaDBDoDKOzgnD)
![{\displaystyle \parallel A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\parallel ={\sqrt {A_{11}^{2}+A_{22}^{2}+A_{33}^{2}+2A_{12}^{2}+2A_{13}^{2}+2A_{23}^{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84yqo3yqhDo2a2zNmQz2eNnjrEnqwQnAdAotw1ajdBntGNyjKOa2zE)
Definition
![{\displaystyle \mathbf {A} :\quad \mathbf {A} =\mathbf {A} ^{\top }\quad {\text{und}}\quad {\vec {v}}\cdot \mathbf {A} \cdot {\vec {v}}>0\quad \forall \;{\vec {v}}\in \mathbb {V} \setminus \{{\vec {0}}\}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AntiOzgaPnjBFyjlFaDJFzjmOots0ytG3nDm5zqo5ajC4oNK3atFC)
Kofaktor:
Mit den #Eigenwerten
, den #Eigenvektoren
und einer reellwertigen Funktion
eines reellen Argumentes
definiert man über das #Eigensystem symmetrischer Tensoren
![{\displaystyle {\begin{aligned}\mathbf {A} =&\sum _{i=1}^{3}\lambda _{i}{\hat {a}}_{i}\otimes {\hat {a}}_{i}\\=&{\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}\cdot {\begin{pmatrix}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&\lambda _{3}\end{pmatrix}}\cdot {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}^{\top }\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NntwNz2w1yqsNyto5ygo2oNG0ata1ota2zAzEngaQntm0zge5aAo1)
den Funktionswert des Tensors:
![{\displaystyle {\begin{aligned}f(\mathbf {A} ):=&\sum _{i=1}^{3}f(\lambda _{i}){\hat {a}}_{i}\otimes {\hat {a}}_{i}\\=&{\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}\cdot {\begin{pmatrix}f(\lambda _{1})&0&0\\0&f(\lambda _{2})&0\\0&0&f(\lambda _{3})\end{pmatrix}}\cdot {\begin{pmatrix}{\hat {a}}_{1}&{\hat {a}}_{2}&{\hat {a}}_{3}\end{pmatrix}}^{\top }\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DztmOaDiPztvFagdEzNsNngnCaAhAygs5zNdFaDa4ajoNaqzDaDK1)
Ist f eine mehrdeutige Funktion, wie die Wurzel (Mathematik), mit n alternativen Werten, dann steht f(A) mehrdeutig für n3 alternative Tensoren.
Insbesondere mit dem Deformationsgradient F:
Rechter Strecktensor
![{\displaystyle \mathbf {U} =+{\sqrt {\mathbf {F} ^{\top }\cdot \mathbf {F} }}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FaDs4oDG4zjm3oqwOztrDnDBBatC1zNwOoqnFoAnCajeOothDaNC1)
Linker Strecktensor
![{\displaystyle \mathbf {v} =+{\sqrt {\mathbf {F\cdot F} ^{\top }}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Oage4nDm3o2nAntoOnDe4z2rFzDs2njiOoDGQz2aNotw1yjsNaNhD)
Henky-Dehnung
![{\displaystyle \mathbf {E} _{H}:=\ln(\mathbf {U} )={\frac {1}{2}}\ln(\mathbf {F} ^{\top }\cdot \mathbf {F} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzqaPzDzBzNoNajhDzqePnDw4oqaNnDw2nji4ytCOagiQaArCngiP)
Die Tensoren
![{\displaystyle {\begin{aligned}\mathbf {S} _{1}=&{\vec {e}}_{1}\otimes {\vec {e}}_{1}\\\mathbf {S} _{2}=&{\vec {e}}_{2}\otimes {\vec {e}}_{2}\\\mathbf {S} _{3}=&{\vec {e}}_{3}\otimes {\vec {e}}_{3}\\\mathbf {S} _{4}=&{\vec {e}}_{2}\otimes {\vec {e}}_{3}+{\vec {e}}_{3}\otimes {\vec {e}}_{2}\\\mathbf {S} _{5}=&{\vec {e}}_{1}\otimes {\vec {e}}_{3}+{\vec {e}}_{3}\otimes {\vec {e}}_{1}\\\mathbf {S} _{6}=&{\vec {e}}_{1}\otimes {\vec {e}}_{2}+{\vec {e}}_{2}\otimes {\vec {e}}_{1}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82otJDoDePyqnBaNa3zja4oDBEnjw3zNFDaDo0ajw3aqe5yqhCotFE)
bilden eine Basis im Vektorraum
der symmetrischen Tensoren zweiter Stufe. Bezüglich dieser Basis können alle symmetrischen Tensoren zweiter Stufe in Voigt'scher Notation dargestellt werden:
![{\displaystyle \mathbf {A} \in \mathrm {sym} (\mathbb {V} ,\mathbb {V} )\quad \rightarrow \quad \mathbf {A} =A_{r}\mathbf {S} _{r}{\hat {=}}{\begin{bmatrix}A_{1}\\A_{2}\\A_{3}\\A_{4}\\A_{5}\\A_{6}\end{bmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AoNK0agrCatw5ytwQathDzjnCzjBAaqo4yjG3aNdFyjs0aqw0o2o5)
Diese Vektoren dürfen addiert, subtrahiert und mit einem Skalar multipliziert werden. Beim Skalarprodukt muss
![{\displaystyle \mathbf {A} :\mathbf {B} =A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}+2A_{4}B_{4}+2A_{5}B_{5}+2A_{6}B_{6}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83ati5aNmNoqw3zAi0z2vCa2vCngs2ajBBnDe5zqaOnqi4zjGQags5)
berücksichtigt werden. Siehe auch #Voigt'sche Notation von Tensoren vierter Stufe.
Definition
![{\displaystyle \mathbf {A} :\quad \mathbf {A} =-\mathbf {A} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BntFAytoOzDFFnDhDzgo4zDvBajFBzNG2aNa3otCPyjzAajlEagi1)
Kofaktor:
#Invarianten:
![{\displaystyle \mathrm {Sp} (\mathbf {A} )=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OytGNzNiPnjw1otC2zDoOytFAaNnEnDK5njrDnDePatvFaAo3aDlB)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} )=-{\frac {1}{2}}\mathrm {Sp} (\mathbf {A} ^{2})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OntC2ntsPztrAyqnDntFAaqwOzNsNnjzAzjrDnDJEnjw0zqi0ngaP)
![{\displaystyle \mathrm {det} (\mathbf {A} )=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Doqo3nDC5aDC4ztlAz2w1njzFyjCOzDJBnAe4ythDyqeOyjaOaDa1)
![{\displaystyle \quad \parallel \mathbf {A} \parallel ={\sqrt {2\mathrm {I} _{2}(\mathbf {A} )}}={\sqrt {-\mathrm {Sp} (\mathbf {A} ^{2})}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zNdCntiQotBFaDa1ngsPythFzqrEnjrCnjrDnte1aqo2yqo5aDdA)
In kartesischen Koordinaten:
![{\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\begin{pmatrix}0&A_{12}&A_{13}\\-A_{12}&0&A_{23}\\-A_{13}&-A_{23}&0\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AyjhEzjC3z2rEnjJBatdBnqaPyjhCa2sOoDnDaAo4zDs5atnAotw2)
#Invarianten:
![{\displaystyle \mathrm {Sp} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ao2i0ngrBntoNntK4aNnEytaPngiQaDFBagsPatCOyja0oDm0aAiQ)
![{\displaystyle \mathrm {I} _{2}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=A_{12}^{2}+A_{13}^{2}+A_{23}^{2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OaNwNoAe3aNo3agi4o2e5njmPatm1ytm5zDJFzNhBate2aNi3oDoQ)
![{\displaystyle \mathrm {det} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OntaQoNJDaDe5ajBDzjo2oDKQnDC1nDoOaNBBnAs0ztaOa2i5zNdD)
![{\displaystyle \parallel A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\parallel ={\sqrt {2}}{\sqrt {A_{12}^{2}+A_{13}^{2}+A_{23}^{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83nAo5yti2ato1a2w2ntC2aDBDz2nDzta4oAoPntzDzAoPnjm5njGN)
Bilinearform:
![{\displaystyle {\vec {u}}\cdot \mathbf {A} \cdot {\vec {v}}=-{\vec {v}}\cdot \mathbf {A} \cdot {\vec {u}}\quad \forall {\vec {u}},{\vec {v}}\in \mathbb {V} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DnjoQoDdCnAoQzjlEzNwQyqsQnDC4nDC2nts4oAw4ngnFytFAoqhF)
![{\displaystyle {\vec {v}}\cdot \mathbf {A} \cdot {\vec {v}}=0\quad \forall {\vec {v}}\in \mathbb {V} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ztvDnqvDats2zNrBzga2o2zAngzAoDsQnjoOzDGOaArDoNdFztBE)
Ein Eigenwert ist null, zwei imaginär konjugiert komplex, siehe #Axialer Tensor oder Kreuzproduktmatrix.
#Dualer axialer Vektor:
![{\displaystyle {\begin{aligned}&\mathbf {A} _{\times }:={\stackrel {A}{\overrightarrow {\mathbf {A} }}}:=-{\frac {1}{2}}\mathbf {1} \times \mathbf {A} ^{\top }=-{\frac {1}{2}}\mathbf {1} \cdot \!\!\times \mathbf {A} =-{\frac {1}{2}}{\vec {\mathrm {i} }}(\mathbf {A} )\\&\rightarrow \quad \mathbf {A} \cdot {\vec {v}}={\stackrel {A}{\overrightarrow {\mathbf {A} }}}\times {\vec {v}}\quad \forall {\vec {v}}\in \mathbb {V} \end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QatnCoNeNytFEoNrDoNvEnjsPa2s2ygrEz2rBaqw0ntwNzNC2oNa5)
mit #Vektorinvariante
. Der zum Eigenwert null gehörende #Eigenvektor ist proportional zum dualen axialen Vektor
denn
![{\displaystyle \mathbf {A\cdot A} _{\times }=\mathbf {A} _{\times }\times \mathbf {A} _{\times }={\vec {0}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DzAvAyje4aDG3ztBBatCOo2dAzDrBatnBnqo0njK5nDe3zNrDzNvF)
![{\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\quad \rightarrow \;\mathbf {A} _{\times }=-{\frac {1}{2}}A_{ij}{\hat {e}}_{i}\times {\hat {e}}_{j}={\begin{pmatrix}-A_{23}\\A_{13}\\-A_{12}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PyqoNnjsPyjrCzNw1zDw0oNBAoNoPzgiQyje0zts3zte5nDo3ntK2)
![{\displaystyle \mathbf {A} =A_{ij}({\vec {a}}_{i}\otimes {\vec {b}}_{j}-{\vec {b}}_{j}\otimes {\vec {a}}_{i})\quad \rightarrow \;\mathbf {A} _{\times }=-A_{ij}{\vec {a}}_{i}\times {\vec {b}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BajwNaqrFa2vBnjnEaAaNz2rAaDlEoDrBnjFDzDJCo2dCaDzDnte1)
Kreuzproduktmatrix
eines Vektors
:
![{\displaystyle {\begin{aligned}{\vec {u}}=u_{i}{\hat {e}}_{i}=&{\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\end{pmatrix}}\\\rightarrow \;[{\vec {u}}]_{\times }=&{\vec {u}}\times \mathbf {1} ={\vec {u}}\times {\hat {e}}_{i}\otimes {\hat {e}}_{i}=-{\stackrel {3}{\mathbf {E} }}\cdot {\vec {u}}={\begin{pmatrix}0&-u_{3}&u_{2}\\u_{3}&0&-u_{1}\\-u_{2}&u_{1}&0\end{pmatrix}}\in {\mathcal {L}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzNdBaNe5nga5zgwOaDvBzjhEoAw2nge2njm5zNvCzDvCatBAoNvF)
Kofaktor:
#Invarianten:
![{\displaystyle \mathrm {Sp} =0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82ngvBajhAyjGNo2o3oqi0oNiPaAs3zjGNaqrCz2hCajGQz2eOnDsO)
![{\displaystyle \mathrm {I} _{2}={\vec {u}}\cdot {\vec {u}}=u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AytePoNrEyqwOaga1a2iPaNCQzgsPnjhBoNw3agvCoNwQnDFAngvC)
![{\displaystyle \mathrm {det} =0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DygwPzNzCz2e0atm5otvFatiNztBBaDhFoDnEzteOzjhCaqi0ngnD)
![{\displaystyle \|{\vec {u}}\times \mathbf {1} \|={\sqrt {2{\vec {u}}\cdot {\vec {u}}}}={\sqrt {2}}{\sqrt {u_{1}^{2}+u_{2}^{2}+u_{3}^{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NotK2ntzDnArDato0njBEzjlFaDGPoNs3yjBDzNC4aNs2agePnDiN)
![{\displaystyle {\stackrel {A}{\overrightarrow {{\vec {u}}\times \mathbf {1} }}}={\vec {u}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zgw0nqrBztoPoqiPage3nte4atdEzNFDaDwNaDzDnDa0njCPnqiP)
#Eigensystem:
![{\displaystyle {\begin{aligned}\lambda _{1}=&0\,,&{\vec {v}}_{1}=&{\vec {u}}\\\lambda _{2,3}=&\mp \mathrm {i} |{\vec {u}}|\,,&{\vec {v}}_{2,3}&\simeq &{\frac {u_{1}}{|{\vec {u}}|}}{\begin{pmatrix}u_{1}\\u_{2}\\u_{3}\end{pmatrix}}\pm \mathrm {i} {\begin{pmatrix}\pm \mathrm {i} |{\vec {u}}|\\-u_{3}\\u_{2}\end{pmatrix}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Nzjw1nDm1oDG4ajm1zja3ygwNoDC0nDm3zjw2oqw2nqeNnjrDnqdB)
Eigenschaften:
![{\displaystyle {\vec {u}}\times {\vec {v}}=({\vec {u}}\times \mathbf {1} )\cdot {\vec {v}}={\vec {u}}\cdot ({\vec {v}}\times \mathbf {1} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaNw5aNrDzjsOngw4nDFAntlEagw4zjdCnjnAo2iNntaQaAwNa2aO)
![{\displaystyle {\vec {u}}\times \mathbf {1} =\mathbf {1} \times {\vec {u}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81otK1zqzBygeQzqvDnAs1o2vFnDBBnte1nDFFztGNnqrDzDvEyga3)
![{\displaystyle ({\vec {u}}\times \mathbf {1} )^{\top }=-{\vec {u}}\times \mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82njaNagi5aNFEoDBAzthCa2iPyqe1zqiPaqnFzDe0yjwNatFAztCN)
![{\displaystyle {\vec {u}}=-{\frac {1}{2}}\mathbf {1} \cdot \!\!\times ({\vec {u}}\times \mathbf {1} )=-{\frac {1}{2}}(\mathbf {1} \times {\vec {u}})\times \mathbf {1} ={\frac {1}{2}}\mathbf {1} \times ({\vec {u}}\times \mathbf {1} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DntdAzqdBnqi3otw2oqhDo2o2zNC4njo2ztFBajnEaNG4nts2aAa0)
![{\displaystyle {\vec {u}}\times ({\vec {v}}\times \mathbf {1} )=({\vec {u}}\times \mathbf {1} )\cdot ({\vec {v}}\times \mathbf {1} )={\vec {v}}\otimes {\vec {u}}-({\vec {u}}\cdot {\vec {v}})\mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zNGQaqw4oqzFzjCNytK1oAhAaAi5yjGOajmQz2e2nqvAoNoOzjKO)
![{\displaystyle {\vec {u}}\times ({\vec {v}}\times \mathbf {1} )\cdot {\vec {w}}={\vec {u}}\times ({\vec {v}}\times {\vec {w}})=({\vec {u}}\cdot {\vec {w}}){\vec {v}}-({\vec {u}}\cdot {\vec {v}}){\vec {w}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AoqdFotKOyqaQzAe5njG0yte1ots0ngoPzAnCoDi2yjdCnAdCajC4)
Potenzen von
![{\displaystyle [{\vec {u}}]_{\times }^{2}=[{\vec {u}}]_{\times }\cdot [{\vec {u}}]_{\times }={\vec {u}}\otimes {\vec {u}}-({\vec {u}}\cdot {\vec {u}})\mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Aatw5ajzEaDvBa2s0zgw0ztJBzNaNytvDntJEzAhBnjiNz2e4zDnB)
![{\displaystyle [{\vec {u}}]_{\times }^{3}=-({\vec {u}}\cdot {\vec {u}})[{\vec {u}}]_{\times }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AatmOoDG1oAhDo2zCatG5nAnFz2w0zqsOaNi1atnAzgwNzDFAzAvD)
Definition
![{\displaystyle \mathbf {A} :\quad \mathrm {Sp} (\mathbf {A} )=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pnto3ythCnAdCzNo2ygdAaji4aqzAzje3ajK0oqrDzNnAzDdEzjaO)
Kofaktor:
#Hauptinvarianten:
![{\displaystyle \mathrm {Sp} (\mathbf {A} ):=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ntzBoqsNoDdAyjJDzjJCotmQoNlDnjK1nja1ajG4nqa4nAe1aqs4)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} )=-{\frac {1}{2}}\mathrm {Sp} (\mathbf {A} ^{2})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OntC2ntsPztrAyqnDntFAaqwOzNsNnjzAzjrDnDJEnjw0zqi0ngaP)
![{\displaystyle \mathrm {det} (\mathbf {A} )={\frac {1}{3}}\mathrm {Sp} (\mathbf {A} ^{3})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EyqoPaNKNatePaNJAaAvFnts4aDC5atlDnjhFzDvDaNG3o2sQnDwN)
Bezüglich der Standardbasis:
![{\displaystyle \mathbf {A} =A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}={\begin{pmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&-A_{11}-A_{22}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzAa4atvFoDK1z2aQnDC5aDvFzjBCagzFataPato4aNJEaqwOytm3)
![{\displaystyle \mathrm {Sp} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ao2i0ngrBntoNntK4aNnEytaPngiQaDFBagsPatCOyja0oDm0aAiQ)
![{\displaystyle \mathrm {I} _{2}(A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=-A_{11}^{2}-A_{22}^{2}-A_{11}A_{22}-A_{12}A_{21}-A_{13}A_{31}-A_{23}A_{32}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DnDsPzjC0njiPoNK4ztJBzDlAaNCOoqvFajFDzqwQaDeNoNdBaNs4)
![{\displaystyle {\begin{aligned}\mathrm {det} (A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j})=&-A_{11}(A_{11}A_{22}+A_{22}^{2}+A_{23}A_{32})\\&+A_{12}(A_{23}A_{31}+A_{21}A_{11}+A_{21}A_{22})\\&+A_{13}(A_{21}A_{32}-A_{22}A_{31})\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fyti3zjGPaNi4oDw1zDnDatw1agi4ntdDntC1aNoNajFEoNhAyqsQ)
![{\displaystyle \parallel A_{ij}{\hat {e}}_{i}\otimes {\hat {e}}_{j}\parallel ={\sqrt {\begin{array}{r}2A_{11}^{2}+2A_{22}^{2}+2A_{11}A_{22}+A_{12}^{2}+A_{21}^{2}+\ldots \\\ldots +A_{13}^{2}+A_{31}^{2}+A_{23}^{2}+A_{32}^{2}\end{array}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82ata0nts3a2hFoNdCnthBnqeOajGQztaNoDhEaNaQaDG0zgaNaDi3)
Definition
![{\displaystyle \mathbf {A} :\quad \mathbf {A} =a\mathbf {1} ={\begin{pmatrix}a&0&0\\0&a&0\\0&0&a\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81nDa4zjnFzqiQngdDyqo4oNe2nAvByqhCnjs1aqvBatGQnDe4yjo3)
Kofaktor:
![{\displaystyle \mathrm {Sp} (\mathbf {A} )=3a}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85yjBFzNsOoDvEythEnqe5ygi4oDlDntBCaDaPzqiPnjBCa2s5nqiP)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} )=3a^{2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85a2aOztsQzto3a2i2z2aQatCOoDoNoNG4oNGQnjw5yjrAotm3z2vC)
![{\displaystyle \mathrm {det} (\mathbf {A} )=a^{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PztaNajsOoqdBzjK4aNvEaNG5zNsNnDdFzAzAyqo3aNG1zqoNyge1)
![{\displaystyle \parallel \mathbf {A} \parallel ={\sqrt {3}}|a|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aNG5zqe1zjo1otC4oDw3aDmNzjeQo2vCyjm1njhCzDnAnqi4zNm2)
Gegeben ein beliebiger Tensor
![{\displaystyle \mathbf {A} ^{\mathrm {S} }=\mathrm {sym} (\mathbf {A} ):={\frac {1}{2}}(\mathbf {A} +\mathbf {A} ^{\top })}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PzAa0oAo0aDs3ote3zgrAzNdFzDw0a2zFoqiQoqsOytKOyqo2aAw5)
![{\displaystyle \mathbf {A} ^{\mathrm {S} }={\frac {1}{2}}{\begin{pmatrix}2A_{11}&A_{12}+A_{21}&A_{13}+A_{31}\\A_{12}+A_{21}&2A_{22}&A_{23}+A_{32}\\A_{13}+A_{31}&A_{23}+A_{32}&2A_{33}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qajm3a2a0ztG4zNK0nqwOnDCOatC3zgnEzgw3zjm4zDFEygvEzgi3)
![{\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {S} })=\mathrm {Sp} (\mathbf {A} )=A_{11}+A_{22}+A_{33}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ytsNatvAytm5aDoOoDmOnAs2o2wNnAo1ytFDa2dCoNBFz2w4oDm4)
![{\displaystyle {\begin{aligned}\mathrm {I} _{2}(\mathbf {A} ^{\mathrm {S} })=&{\frac {1}{2}}\mathrm {I} _{2}(\mathbf {A} )+{\frac {1}{4}}\mathrm {Sp} ^{2}(\mathbf {A} )-{\frac {1}{4}}\mathbf {A:A} \\=&A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}\\&-{\frac {1}{4}}\left[(A_{12}+A_{21})^{2}+(A_{13}+A_{31})^{2}+(A_{23}+A_{32})^{2}\right]\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80oDvEyjJBnjvAaAsPytC5yqzEnjlEytiNnjCOzjFBzNGQzqs1othA)
![{\displaystyle {\begin{aligned}\mathrm {det} (\mathbf {A} ^{\mathrm {S} })=&{\frac {1}{4}}\mathrm {det} (\mathbf {A} )+{\frac {1}{4}}\mathbf {A} :\mathrm {adj} (\mathbf {A} )\\=&A_{11}A_{22}A_{33}+{\frac {1}{4}}(A_{12}+A_{21})(A_{23}+A_{32})(A_{13}+A_{31})\\&-{\frac {1}{4}}\left[A_{11}(A_{23}+A_{32})^{2}+A_{22}(A_{13}+A_{31})^{2}+A_{33}(A_{12}+A_{21})^{2}\right]\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DntmPnAnEnAsNoAe5zDzCatm3zji1nDe1zje5nqi1njC4nDC5aDvD)
![{\displaystyle {\begin{aligned}\parallel (\mathbf {A} ^{\mathrm {S} })\parallel =&{\sqrt {\mathbf {A:A} ^{\mathrm {S} }}}\\=&{\sqrt {A_{11}^{2}+A_{22}^{2}+A_{33}^{2}+{\frac {1}{2}}[(A_{12}+A_{21})^{2}+(A_{13}+A_{31})^{2}+(A_{23}+A_{32})^{2}]}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aNi0aNCPaAhCo2sQotwQz2i3zje0z2rDatvFaqi1aNa1yjCNytmO)
![{\displaystyle \mathbf {A} ^{\mathrm {A} }=\mathrm {skw} (\mathbf {A} ):={\frac {1}{2}}(\mathbf {A} -\mathbf {A} ^{\top })}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AoNhEnjo0zDs1aNoPnjo0zta0aqaOntlCoDnCatCPaNK2ntm2z2vD)
![{\displaystyle \mathbf {A} ^{\mathrm {A} }={\frac {1}{2}}{\begin{pmatrix}0&A_{12}-A_{21}&A_{13}-A_{31}\\A_{21}-A_{12}&0&A_{23}-A_{32}\\A_{31}-A_{13}&A_{32}-A_{23}&0\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Oo2oQytzCotCQyqa2ntG0aqo3agdBzNJFaNaPnDoPygrEaAvBaAeQ)
![{\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {A} })=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83otoOyqw1aNiOaNBCaqrCnjrDo2s3aNdDajFCzDeOzAa1zDnCzjw4)
![{\displaystyle {\begin{aligned}\mathrm {I} _{2}(\mathbf {A} ^{\mathrm {A} })=&{\frac {1}{4}}\left[\mathbf {A:A} -\mathrm {Sp} (\mathbf {A} ^{2})\right]\\=&{\frac {1}{4}}\left[(A_{12}-A_{21})^{2}+(A_{13}-A_{31})^{2}+(A_{23}-A_{32})^{2}\right]\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AageNajBBoNFDats3aqzAzqzDyjKNzNiQzDdBoqs4ztm3oAa0zga1)
![{\displaystyle \mathrm {det} (\mathbf {A} ^{\mathrm {A} })=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaDK0oDo5nAzCo2dFaAnBo2a4njm2o2o0ajvFo2dFo2dAnAvAygiP)
![{\displaystyle \parallel \mathbf {A} ^{\mathrm {A} }\parallel ={\sqrt {\mathbf {A:A} ^{\mathrm {A} }}}={\sqrt {\frac {1}{2}}}{\sqrt {(A_{12}-A_{21})^{2}+(A_{13}-A_{31})^{2}+(A_{32}-A_{23})^{2}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnjnCaqePntnDygzEati4zNnFo2dFzNBDagdBz2o2a2s0zNFBo2dF)
![{\displaystyle \mathbf {A} ^{\mathrm {D} }=\mathrm {dev} (\mathbf {A} ):=\mathbf {A} -{\frac {1}{3}}\mathrm {Sp} (\mathbf {A} )\mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zDK3agwQoNhEoAhCo2a3oqzCzgiQaDs0ngzBztK5oDi1o2e0agoP)
![{\displaystyle \mathbf {A} ^{\mathrm {D} }={\begin{pmatrix}{\frac {2A_{11}-A_{22}-A_{33}}{3}}&A_{12}&A_{13}\\A_{21}&{\frac {2A_{22}-A_{11}-A_{33}}{3}}&A_{23}\\A_{31}&A_{32}&{\frac {2A_{33}-A_{11}-A_{22}}{3}}\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ntG5njnCoqrDzjzAoDzEaDaQnAdBytmPyjJDzDwNnDsPaAw1ztoN)
![{\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {D} })=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81njC2oNC4ytFEzqsOaqi5zqa0atiPaNeNzDvCago4ajdEa2wQaDm5)
![{\displaystyle {\begin{aligned}\mathrm {I} _{2}(\mathbf {A} ^{\mathrm {D} })=&\mathrm {I} _{2}(\mathbf {A} )-{\frac {1}{3}}\mathrm {Sp} ^{2}(\mathbf {A} )={\frac {1}{6}}\mathrm {Sp} ^{2}(\mathbf {A} )-{\frac {1}{2}}\mathrm {Sp} (\mathbf {A} ^{2})\\=&{\frac {1}{3}}(A_{11}A_{22}+A_{11}A_{33}+A_{22}A_{33}-A_{11}^{2}-A_{22}^{2}-A_{33}^{2})\\&-A_{12}A_{21}-A_{13}A_{31}-A_{23}A_{32}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QoDBBzqe4oDKQaDC3atK0aNaPoNrBoNi3oqhByjGOnDK3otFBaDC3)
![{\displaystyle {\begin{aligned}\mathrm {det} (\mathbf {A} ^{\mathrm {D} })=&\mathrm {det} (\mathbf {A} )+{\frac {2}{27}}\mathrm {Sp} ^{3}(\mathbf {A} )-{\frac {1}{3}}\mathrm {Sp} (\mathbf {A} )\mathrm {I} _{2}(\mathbf {A} )\\=&{\frac {1}{27}}{\Big [}12A_{11}A_{22}A_{33}+2(A_{11}^{3}+A_{22}^{3}+A_{33}^{3})\ldots \\&\qquad \ldots -3A_{11}^{2}(A_{22}+A_{33})-3A_{22}^{2}(A_{11}+A_{33})-3A_{33}^{2}(A_{11}+A_{22}){\Big ]}\\&-{\frac {1}{3}}{\Big [}(2A_{11}-A_{22}-A_{33})A_{23}A_{32}+(2A_{22}-A_{11}-A_{33})A_{13}A_{31}+\ldots \\&\qquad \ldots +(2A_{33}-A_{11}-A_{22})A_{12}A_{21}{\Big ]}\\&+A_{13}A_{32}A_{21}+A_{12}A_{23}A_{31}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85agw5zNKNajm2njhAothCzqvCzgvCnjK0zje3zjK4z2i5zDi5nDK4)
![{\displaystyle \parallel \mathbf {A} ^{\mathrm {D} }\parallel ={\sqrt {\begin{array}{r}{\frac {2}{3}}(A_{11}^{2}+A_{22}^{2}+A_{33}^{2}-A_{11}A_{22}-A_{11}A_{33}-A_{22}A_{33})+\ldots \\\ldots +A_{12}^{2}+A_{21}^{2}+A_{13}^{2}+A_{31}^{2}+A_{23}^{2}+A_{32}^{2}\end{array}}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QajnCoqaOntCPzDKNntm4zti3o2i2a2o5nAo2ngeOoqdCngePaNm4)
![{\displaystyle \mathbf {A} ^{\mathrm {K} }=\mathrm {sph} (\mathbf {A} ):={\frac {1}{3}}\mathrm {Sp} (\mathbf {A} )\mathbf {1} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zjlAyghAzNnCzjaPnjzAngi5nqrEo2dBotzFaDC5aNC4njnCaAe3)
![{\displaystyle \mathbf {A} ^{\mathrm {K} }={\frac {1}{3}}(A_{11}+A_{22}+A_{33}){\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ata2zNzFo2zAytlBaDvFagoQoDo4yjGNyjw1oNw3atJCnDFCoqe1)
![{\displaystyle \mathrm {Sp} (\mathbf {A} ^{\mathrm {K} })=\mathrm {Sp} (\mathbf {A} )=A_{11}+A_{22}+A_{33}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BzDwNyto2zDvCo2nAnqsOaNJAaNhBo2a5ygiOoDzByqs0otm4yto1)
![{\displaystyle \mathrm {I} _{2}(\mathbf {A} ^{\mathrm {K} })={\frac {1}{3}}\mathrm {Sp} ^{2}(\mathbf {A} )={\frac {1}{3}}(A_{11}+A_{22}+A_{33})^{2}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85oAzBzNdBatC0zDmOyjnBygs0ytK3zjs2yte3zArCzje4nDdDoNK5)
![{\displaystyle \mathrm {det} (\mathbf {A} ^{\mathrm {K} })={\frac {1}{27}}\mathrm {Sp} ^{3}(\mathbf {A} )={\frac {1}{27}}(A_{11}+A_{22}+A_{33})^{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FaDwNaAdCnAwQajoNzDrFyteQzDe3nDe1ntFCoDwNoDw5aqrEzNJE)
![{\displaystyle \parallel \mathbf {A} ^{\mathrm {K} }\parallel ={\frac {1}{\sqrt {3}}}|\mathrm {Sp} (\mathbf {A} )|={\frac {1}{\sqrt {3}}}|A_{11}+A_{22}+A_{33}|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85nqa1nja3nAs0atCQats0njm0zDa5otaNoNC5o2i3zDmNytBEoNBE)
![{\displaystyle \mathbf {A} =\mathbf {A} ^{\mathrm {S} }+\mathbf {A} ^{\mathrm {A} }=\mathbf {A} ^{\mathrm {D} }+\mathbf {A} ^{\mathrm {K} }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qoqs5oDCQaghFytwQnAnByts3oDrEyqa2atsOaDw0ytKPaghBygw3)
Symmetrische und schiefsymmetrische Tensoren sind orthogonal zueinander:
![{\displaystyle \mathbf {A} ^{\mathrm {S} }:\mathbf {B} ^{\mathrm {A} }=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BnDKNoNdEaga0nqi4nDlFoqs1ntiPaAo1aje5o2w2ztdCzje5zgo1)
Deviatoren und Kugeltensoren sind orthogonal zueinander:
![{\displaystyle \mathbf {A} ^{\mathrm {D} }:\mathbf {B} ^{\mathrm {K} }=0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DoDCNo2nCnDlFyjCQotKPotrEotFDoqe5zja2oNw1oDJCyjeNagoP)
Für jeden Tensor F mit #Determinante ≠ 0 gibt es #Orthogonale Tensoren Q und #Symmetrische und positiv definite Tensoren U in eindeutiger Weise, sodass
- F = Q·U
Im Fall des Deformationsgradienten ist U der rechte Strecktensor, siehe #Symmetrische und positiv definite Tensoren. Der Anteil U berechnet sich wie dort angegeben aus
![{\displaystyle \mathbf {U} =+{\sqrt {\mathbf {F^{\top }\cdot F} }}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82nqwOoNaNate4zDhAzgiQoDCQatlEo2aQatJDate5aNa4oNnCnjBD)
Dann ist U·U = F⊤·F und
![{\displaystyle \mathbf {Q} =\mathbf {F\cdot U} ^{-1}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ytCNnDJBoqo0oNK4nDrEnAzDathFoDrDoAwOaAsPyqw2a2a1zNBE)
Bei det(F)=0 ergeben sich U sowie Q aus der Singulärwertzerlegung von F und U ist nur noch symmetrisch positiv semidefinit.
Gegeben sei die Gerade durch den Punkt
mit Richtungsvektor
und ein beliebiger anderer Punkt
.
Dann ist
![{\displaystyle {\begin{aligned}{\vec {p}}=&{\vec {x}}+{\vec {a}}+{\vec {b}}\quad {\textsf {mit}}\quad {\vec {a}}\|{\vec {g}}\quad {\text{und}}\quad {\vec {b}}\bot {\vec {g}}\\\mathbf {G} =&{\frac {{\vec {g}}\otimes {\vec {g}}}{{\vec {g}}\cdot {\vec {g}}}}\quad \rightarrow \quad \mathbf {G} \cdot {\vec {g}}={\vec {g}}\,,\quad (\mathbf {1} -\mathbf {G} )\cdot {\vec {g}}={\vec {0}}\\&{\vec {n}}\cdot {\vec {g}}=0\quad \rightarrow \quad \mathbf {G} \cdot {\vec {n}}={\vec {0}}\,,\quad (\mathbf {1} -\mathbf {G} )\cdot {\vec {n}}={\vec {n}}\\{\vec {a}}=&\mathbf {G} \cdot ({\vec {p}}-{\vec {x}})={\frac {{\vec {g}}\cdot ({\vec {p}}-{\vec {x}})}{{\vec {g}}\cdot {\vec {g}}}}{\vec {g}}\\{\vec {b}}=&\left(\mathbf {1} -\mathbf {G} \right)\cdot ({\vec {p}}-{\vec {x}})={\vec {p}}-{\vec {x}}-{\vec {a}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FoAa4zDo3aDiOnje3ytw4zqiPoDa1zgo0ntvCaNiQntwPzNwPzAs3)
Der Punkt
ist die senkrechte Projektion von
auf die Gerade. Der Tensor G extrahiert den Anteil eines Vektors in Richtung von
und 1-G den Anteil senkrecht dazu.
Gegeben sei die Ebene durch den Punkt
und zwei die Ebene aufspannende Vektoren
und
sowie ein beliebiger anderer Punkt
. Dann verschwindet die Normale
![{\displaystyle {\hat {n}}={\frac {{\vec {u}}\times {\vec {v}}}{|{\vec {u}}\times {\vec {v}}|}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FnAiQnDrFnqrDzjBCzti4agoQzNm4aNo0zDs5yjvAytJBnjvFzNG1)
nicht. Dann ist
![{\displaystyle {\begin{aligned}{\vec {p}}=&{\vec {x}}+{\vec {a}}+{\vec {b}}\quad {\textsf {mit}}\quad {\vec {a}}\bot {\hat {n}}\quad {\text{und}}\quad {\vec {b}}\|{\hat {n}}\\\mathbf {P} =&{\frac {({\vec {v}}\cdot {\vec {v}}){\vec {u}}\otimes {\vec {u}}-({\vec {u}}\cdot {\vec {v}})({\vec {u}}\otimes {\vec {v}}+{\vec {v}}\otimes {\vec {u}})+({\vec {u}}\cdot {\vec {u}}){\vec {v}}\otimes {\vec {v}}}{({\vec {u}}\cdot {\vec {u}})({\vec {v}}\cdot {\vec {v}})-({\vec {u}}\cdot {\vec {v}})^{2}}}=\mathbf {1} -{\hat {n}}\otimes {\hat {n}}\\&\rightarrow \mathbf {P} \cdot {\vec {u}}={\vec {u}}\,,\quad \mathbf {P} \cdot {\vec {v}}={\vec {v}}\,,\quad \mathbf {P} \cdot {\hat {n}}={\vec {0}}\,,\quad (\mathbf {1} -\mathbf {P} )\cdot {\hat {n}}={\hat {n}}\\&\rightarrow \mathbf {P} \cdot (x{\vec {u}}+y{\vec {v}})=x{\vec {u}}+y{\vec {v}}\quad {\text{und}}\quad (\mathbf {1} -\mathbf {P} )\cdot (x{\vec {u}}+y{\vec {v}})={\vec {0}}\quad \forall x,y\in \mathbb {R} \\{\vec {a}}=&\mathbf {P} \cdot ({\vec {p}}-{\vec {x}})\\{\vec {b}}=&(\mathbf {1} -\mathbf {P} )\cdot ({\vec {p}}-{\vec {x}})={\vec {p}}-{\vec {x}}-{\vec {a}}\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NotK4njsNaqoQzAaOztwQyqaPzAnAoti4oNFBoDaOzNi0zge3ztiO)
Der Punkt
ist die senkrechte Projektion von
auf die Ebene.[2] Der Tensor P extrahiert den Anteil eines Vektors in der Ebene und 1-P den Anteil senkrecht dazu.
Die Projektion der Geraden, die durch die Punkte
und
verläuft, liegt in der Ebene in Richtung des Vektors
.
Falls
und
folgt:
![{\displaystyle {\hat {n}}={\vec {u}}\times {\vec {v}}\quad {\text{mit}}\quad |{\hat {n}}|=1}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OaDKQoNnByjK5yjzEnjaQnqa4ytdBzNaQnge4nAaNyjJDyti4aNJF)
![{\displaystyle \mathbf {P} ={\vec {u}}\otimes {\vec {u}}+{\vec {v}}\otimes {\vec {v}}=\mathbf {1} -{\hat {n}}\otimes {\hat {n}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AztmOati3aDCQaNmNzDs2zDe0nqw1zgwQa2zAyjKPnArEzNmQnqzA)
![{\displaystyle {\vec {a}}=({\vec {u}}\otimes {\vec {u}}+{\vec {v}}\otimes {\vec {v}})\cdot ({\vec {p}}-{\vec {x}})=(\mathbf {1} -{\hat {n}}\otimes {\hat {n}})\cdot ({\vec {p}}-{\vec {x}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9ByjeNzjJEzje5yjdDzjm1ngiQnDnCoDJAa2aOyqo0nDiQyti1zDhB)
![{\displaystyle {\vec {b}}=(\mathbf {1} -{\vec {u}}\otimes {\vec {u}}-{\vec {v}}\otimes {\vec {v}})\cdot ({\vec {p}}-{\vec {x}})=({\hat {n}}\otimes {\hat {n}})\cdot ({\vec {p}}-{\vec {x}})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BoNnDaqw0atwNzqe5oNzAatoQnqzBotJCzqdDyjhAygnDytJDoNKQ)
Definition:
![{\displaystyle {\begin{aligned}{\stackrel {3}{\mathbf {E} }}:=&\epsilon _{ijk}\,{\hat {e}}_{i}\otimes {\hat {e}}_{j}\otimes {\hat {e}}_{k}\\=&({\hat {e}}_{j}\times {\hat {e}}_{k})\otimes {\hat {e}}_{j}\otimes {\hat {e}}_{k}\\=&{\hat {e}}_{i}\otimes ({\hat {e}}_{k}\times {\hat {e}}_{i})\otimes {\hat {e}}_{k}\\=&{\hat {e}}_{i}\otimes {\hat {e}}_{j}\otimes ({\hat {e}}_{i}\times {\hat {e}}_{j})\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AotrEoNaQntvDygi5zDi3yjw1ytCPaqw0ago1o2dDaNrAaNCNoNwN)
Kreuzprodukt von Vektoren:
![{\displaystyle {\vec {u}}\times {\vec {v}}={\stackrel {3}{\mathbf {E} }}:({\vec {u}}\otimes {\vec {v}})={\vec {v}}\cdot {\stackrel {3}{\mathbf {E} }}\cdot {\vec {u}}=-{\vec {u}}\cdot {\stackrel {3}{\mathbf {E} }}\cdot {\vec {v}}=-{\stackrel {3}{\mathbf {E} }}:({\vec {v}}\otimes {\vec {u}})=-{\vec {v}}\times {\vec {u}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82njm3njaOoNm3agsPa2aQoNwQatwPzDs1oAzAajzFa2iQatC5zjoN)
![{\displaystyle {\vec {e}}_{i}\times {\vec {e}}_{j}=\epsilon _{ijk}\,{\hat {e}}_{k}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DotrAntFEztwQataPa2i2aNvFyjFFa2zCnAnCo2nEzgs3oAvFoti5)
#Kreuzprodukt von Tensoren, #Skalarkreuzprodukt von Tensoren:
![{\displaystyle {\begin{aligned}{\stackrel {3}{\mathbf {E} }}:\mathbf {A} =&\mathbf {A} :{\stackrel {3}{\mathbf {E} }}=-{\stackrel {3}{\mathbf {E} }}:(\mathbf {A} ^{\top })=-(\mathbf {A} ^{\top }):{\stackrel {3}{\mathbf {E} }}\\=&\mathbf {1} \times \mathbf {A} ^{\top }=\mathbf {1} \cdot \!\!\times \mathbf {A} \end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaDC2zjiNzjC5zjo5yjlDo2vEoAePnDsPaNKOzgnDo2hEoNvDyte2)
#Dualer axialer Vektor und #Vektorinvariante:
![{\displaystyle {\stackrel {3}{\mathbf {E} }}:\mathbf {A} =-2{\stackrel {A}{\overrightarrow {\mathbf {A} }}}={\vec {\mathrm {i} }}(\mathbf {A} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OatG2z2vEoDFEngaQytBDnjsOzjaQzgs3ztK2njJAnqvFatsQntBD)
#Kreuzprodukt von Tensoren:
![{\displaystyle \mathbf {A} \times \mathbf {B} ={\stackrel {3}{\mathbf {E} }}:(\mathbf {A\cdot B} ^{\top })}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Noqw0zDJDnDCPngoOnAiQytC3ztnCoto5zDG1ajdEaAzBoNlFzjnD)
![{\displaystyle (A_{ik}{\vec {e}}_{i}\otimes {\vec {e}}_{k})\times (B_{jl}{\vec {e}}_{j}\otimes {\vec {e}}_{l})=A_{ik}B_{jk}{\vec {e}}_{i}\times {\vec {e}}_{j}=\epsilon _{ijk}A_{jl}B_{kl}{\vec {e}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QzDaQyqaOygeOz2eQatG1nDsOnDhCaDC0zgs3aqeOzjo1zNeOzta5)
#Skalarkreuzprodukt von Tensoren:
![{\displaystyle \mathbf {A} \cdot \!\!\times \mathbf {B} ={\stackrel {3}{\mathbf {E} }}:(\mathbf {A\cdot B} )}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ao2nBzgzAyqhDzNm4zDsQzNm2oNKNzqe2ajGPygdEaji0z2dAzjGN)
![{\displaystyle (A_{ik}{\vec {e}}_{i}\otimes {\vec {e}}_{k})\cdot \!\!\times (B_{lj}{\vec {e}}_{l}\otimes {\vec {e}}_{j})=A_{ik}B_{kj}{\vec {e}}_{i}\times {\vec {e}}_{j}=\epsilon _{ijk}A_{jl}B_{lk}{\vec {e}}_{i}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aNs1ytC1ytnBntw5z2i5nDs1a2iOytsOztzDzji5oDvEnDm5aNwO)
#Axialer Tensor oder Kreuzproduktmatrix:
![{\displaystyle {\stackrel {3}{\mathbf {E} }}\cdot {\vec {u}}={\vec {u}}\cdot {\stackrel {3}{\mathbf {E} }}=-{\vec {u}}\times \mathbf {1} =-\mathbf {1} \times {\vec {u}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzNo5nAdDaqi2yjvFygnByqsQnjGPa2s0otBBygrDzNa5ytoNyjs3)
Tensoren zweiter Stufe sind ebenfalls Elemente eines Vektorraums
wie im Abschnitt #Tensoren als Elemente eines Vektorraumes dargestellt. Daher kann man Tensoren vierter Stufe definieren, indem man in dem Kapitel formal die Tensoren zweiter Stufe durch Tensoren vierter Stufe und die Vektoren durch Tensoren zweiter Stufe ersetzt, z. B.:
![{\displaystyle {\stackrel {4}{\mathbf {A} }}=A_{pq}(\mathbf {A} _{p}\otimes \mathbf {G} _{q})}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83ygdAoqe0oNBDnjhFoDa2zjGQygrAnjm5nAs3otnDa2hFygrFngw3)
mit Komponenten
und die Tensoren
sowie
bilden eine Basis von
.
Standardbasis in
:
![{\displaystyle \mathbf {E} _{1}={\vec {e}}_{1}\otimes {\vec {e}}_{1},\mathbf {E} _{2}={\vec {e}}_{1}\otimes {\vec {e}}_{2},\mathbf {E} _{3}={\vec {e}}_{1}\otimes {\vec {e}}_{3},\mathbf {E} _{4}={\vec {e}}_{2}\otimes {\vec {e}}_{1},\ldots ,\mathbf {E} _{9}={\vec {e}}_{3}\otimes {\vec {e}}_{3}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PatJAzAa0aqvFzDBEzqdFzAoPnAa4zga2yjzFyjBDnqs2ago4ytiP)
Tensortransformation:
![{\displaystyle {\stackrel {4}{\mathbf {A} }}:\mathbf {H} =A_{pq}(\mathbf {A} _{p}\otimes \mathbf {G} _{q}):\mathbf {H} :=A_{pq}(\mathbf {G} _{q}:\mathbf {H} )\mathbf {A} _{p}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Engo2atzBzDBFaNvCoNnAzNBAyjaPyqe4yjw3aDzFaNhCajK1oqhA)
Tensorprodukt:
![{\displaystyle [A_{pq}(\mathbf {A} _{p}\otimes \mathbf {G} _{q})]:[B_{rs}(\mathbf {H} _{r}\otimes \mathbf {U} _{s})]:=A_{pq}(\mathbf {G} _{q}:\mathbf {H} _{r})B_{rs}\mathbf {A} _{p}\otimes \mathbf {U} _{s}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BytsOztKQa2dCngrFnqw3zqvEzjFEnqo0zqnBo2zFats1yjC1aDe0)
Übliche Schreibweisen für Tensoren vierter Stufe:
![{\displaystyle {\stackrel {4}{\mathbf {A} }}=\mathbb {A} =A_{ijkl}\;{\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80a2vBnDaOags0aDK0agiOytdCajFEyteOotaQzNC3nDsPytdFytlA)
Transposition:
![{\displaystyle (\mathbf {A} \otimes \mathbf {B} )^{\top }=\mathbf {B} \otimes \mathbf {A} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OoAe4yta1ajm4oNvBzDiOnAdBzNm0ntCOyjFCzDG4zjnAoNC0zqwO)
![{\displaystyle (A_{ijkl}\;{\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})^{\top }:=A_{ijkl}\;{\vec {e}}_{k}\otimes {\vec {e}}_{l}\otimes {\vec {e}}_{i}\otimes {\vec {e}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fygi2zAi0a2wPyqnAaDFFathBzqePoAnFzNG1oDaPngnDyjaPnqhE)
Spezielle Transposition
vertauscht
-tes mit
-tem Basissystem.
Beispielsweise:
![{\displaystyle {\stackrel {4}{\mathbf {A} }}{}^{\stackrel {13}{\top }}:=A_{ijkl}\;{\vec {e}}_{k}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{i}\otimes {\vec {e}}_{l}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzqvDo2i1ytrBoAdFz2vEoto4a2s5nqw1nto2ajG2ajKPzNzCaNnB)
![{\displaystyle {\stackrel {4}{\mathbf {A} }}{}^{\stackrel {24}{\top }}:=A_{ijkl}\;{\vec {e}}_{i}\otimes {\vec {e}}_{l}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Ezja2z2dBajlDataQatG5aNw2nDBByte5a2vBntePzqdEaqw1ntBC)
![{\displaystyle {\stackrel {4}{\mathbf {A} }}\,^{\top }=\left({\stackrel {4}{\mathbf {A} }}{}^{\stackrel {13}{\top }}\right){}^{\stackrel {24}{\top }}=A_{ijkl}\;{\vec {e}}_{k}\otimes {\vec {e}}_{l}\otimes {\vec {e}}_{i}\otimes {\vec {e}}_{j}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzjlEnAdFzja2zNsOaNBEythDnjhCoqvFzDw3atzBnAa0yta0oDsO)
Definition:
Dann gilt:
![{\displaystyle {\begin{aligned}{\stackrel {4}{\mathbf {1} }}:=&\mathbf {E} _{p}\otimes \mathbf {E} _{p}={\stackrel {4}{\mathbf {1} }}{}^{\top }=(\mathbf {1} \otimes \mathbf {1} )\,^{\stackrel {23}{\top }}\\=&{\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{i}\otimes {\vec {e}}_{j}=\delta _{ik}\delta _{jl}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83yqhCzqsNzDs1otG0zNlBoDwOaDs0aDaOaDsNzNlEzga0aqa2nAo2)
Für beliebige Tensoren zweiter Stufe A gilt:
![{\displaystyle {\stackrel {4}{\mathbf {C} }}=\mathbf {E} _{p}^{\top }\otimes \mathbf {E} _{p}=\delta _{il}\delta _{jk}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {A} =\mathbf {A} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NnAdEo2w2oDzBajdEoAeOngi3yja3a2sQngsQotm5aqiNote1zqhB)
![{\displaystyle {\stackrel {4}{\mathbf {C} }}={\frac {1}{3}}\mathbf {1} \otimes \mathbf {1} ={\frac {1}{3}}\delta _{ij}\delta _{kl}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {A} =\mathbf {A} ^{\mathrm {K} }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NaAsNnti4njCOoArFagw4aNG5otJBnjC1aNJAytC2njrEyqvCzNa3)
![{\displaystyle {\stackrel {4}{\mathbf {C} }}={\stackrel {4}{\mathbf {1} }}-{\frac {1}{3}}\mathbf {1} \otimes \mathbf {1} =(\delta _{ik}\delta _{jl}-{\frac {1}{3}}\delta _{ij}\delta _{kl})({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {A} =\mathbf {A} ^{\mathrm {D} }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81nqoNajvEoNo3otBDa2dDzNKOajs0aji4aNwPzgoPyqwOzNlFo2vE)
![{\displaystyle {\stackrel {4}{\mathbf {C} }}={\frac {1}{2}}\left({\stackrel {4}{\mathbf {1} }}+\mathbf {E} _{p}^{\top }\otimes \mathbf {E} _{p}\right)={\frac {1}{2}}(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk})({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {A} =\mathbf {A} ^{\mathrm {S} }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85ote4aqa5oNFCoNFAnjnDztBCotzDnDs3zAw1zDe0zqdDoNBEaDe0)
![{\displaystyle {\stackrel {4}{\mathbf {C} }}={\frac {1}{2}}\left({\stackrel {4}{\mathbf {1} }}-\mathbf {E} _{p}^{\top }\otimes \mathbf {E} _{p}\right)={\frac {1}{2}}(\delta _{ik}\delta _{jl}-\delta _{il}\delta _{jk})({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {A} =\mathbf {A} ^{\mathrm {A} }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaDCOoAhAaAzBnqdCaqo2nAhDzjC2zDvEatoOnthFnje0zqwQnDw1)
Diese fünf Tensoren sind sämtlich symmetrisch.
Mit beliebigen Tensoren zweiter Stufe A, B und G gilt:
![{\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} \otimes \mathbf {B} ^{\top })^{\stackrel {23}{\top }}=A_{ik}B_{lj}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A\cdot G\cdot B} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnjrCoAi1zDnFnqi4aDe5oNaPygrBajm1zqi3ztvDatoPngiOnthE)
![{\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} ^{\top }\otimes \mathbf {B} ^{\top })^{\stackrel {23}{\top }}=A_{ki}B_{lj}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A} ^{\top }\cdot \mathbf {G\cdot B} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NoteNnDFBnDs1zgoPztKPatwOzAoOoAw4njK4aNCNathAngo0o2s1)
![{\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} \otimes \mathbf {B} )^{\stackrel {23}{\top }}=A_{ik}B_{jl}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A\cdot G\cdot B} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82yjJFyqi5aqnDa2nEntC3aje2a2dCnDJAaDCQagvDzAdBa2a3oDC1)
![{\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} ^{\top }\otimes \mathbf {B} )^{\stackrel {23}{\top }}=A_{ki}B_{jl}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A} ^{\top }\cdot \mathbf {G\cdot B} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AngvAa2vBa2vEageQotoNzDmPygnEzAo2ntdAaghFaNFCatnEaAaN)
In dem in diesen Formeln im Tensor vierter Stufe B durch B⊤ und die Transpositionen
durch
ersetzt werden, entstehen die Ergebnisse mit transponiertem G:
![{\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} \otimes \mathbf {B} )^{\stackrel {24}{\top }}=A_{il}B_{kj}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A\cdot G} ^{\top }\cdot \mathbf {B} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82aNo1aqwQygiPzDGPzAvCngi2aAe4njaNatBBnji0zDeNnAa0oNG3)
![{\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} ^{\top }\otimes \mathbf {B} )^{\stackrel {24}{\top }}=A_{li}B_{kj}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A} ^{\top }\cdot \mathbf {G} ^{\top }\cdot \mathbf {B} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AzjCOntzBzDBCz2vEnAaOoDaQygiOajlFntC4oNoQngiPzDC0ngi0)
![{\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} \otimes \mathbf {B} ^{\top })^{\stackrel {24}{\top }}=A_{il}B_{jk}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A\cdot G} ^{\top }\cdot \mathbf {B} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DoqnBaNdCagaOaNi3nta3ntrFaqoQyts4aDCPo2w0nDi4zgaQaDG2)
![{\displaystyle {\stackrel {4}{\mathbf {C} }}=(\mathbf {A} ^{\top }\otimes \mathbf {B} ^{\top })^{\stackrel {24}{\top }}=A_{li}B_{jk}({\vec {e}}_{i}\otimes {\vec {e}}_{j}\otimes {\vec {e}}_{k}\otimes {\vec {e}}_{l})\rightarrow {\stackrel {4}{\mathbf {C} }}:\mathbf {G} =\mathbf {A} ^{\top }\cdot \mathbf {G} ^{\top }\cdot \mathbf {B} ^{\top }}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80agi5atlBzDs0oDwQoto5ajG4zjo3yqs5zDFFoNw2zNiPoqrDzNa0)
![{\displaystyle \left(a{\stackrel {4}{\mathbf {1} }}+\mathbf {B} \otimes \mathbf {C} \right)^{-1}={\frac {1}{a}}\left({\stackrel {4}{\mathbf {1} }}-{\frac {1}{a+\mathbf {B} :\mathbf {C} }}\mathbf {B} \otimes \mathbf {C} \right)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80aAaPzDs5zjoOnDmOa2a0ytePo2s5yqhDnAw4zDoOnjiQntaNzNK2)
Mit den Spannungen
und den Dehnungen
im Hooke'schen Gesetz gilt:
![{\displaystyle {\stackrel {4}{\mathbf {C} }}:=2\mu {\stackrel {4}{\mathbf {1} }}+\lambda \mathbf {1} \otimes \mathbf {1} \quad \rightarrow \quad {\stackrel {4}{\mathbf {C} }}:\mathbf {E} =\mathbf {T} }](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EzAoQzjFAzAi5athAzAo5ajvCnga2ztm0nDzAnqrDythEoAe4aqeQ)
mit den Lamé-Konstanten
und
. Dieser Elastizitätstensor ist symmetrisch.
Invertierungsformel mit
,
und
:
![{\displaystyle {\begin{aligned}&{\stackrel {4}{\mathbf {S} }}:={\stackrel {4}{\mathbf {C} }}{}^{-1}={\frac {1}{2\mu }}\left({\stackrel {4}{\mathbf {1} }}-{\frac {\lambda }{2\mu +3\lambda }}\mathbf {1} \otimes \mathbf {1} \right)={\frac {1}{2\mu }}{\stackrel {4}{\mathbf {1} }}-{\frac {\nu }{E}}\mathbf {1} \otimes \mathbf {1} \\&\rightarrow \quad {\stackrel {4}{\mathbf {S} }}:\mathbf {T} =\mathbf {E} \end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnqrCatKQoNnEnjzFaDvEygnBzjhDaDi4aja5zAzEaqs1oDC0zDvF)
mit der Querdehnzahl
und dem Elastizitätsmodul
.
Aus der Basis
des Vektorraums
der symmetrischen Tensoren zweiter Stufe, siehe #Voigt-Notation symmetrischer Tensoren zweiter Stufe, kann eine Basis des Vektorraums
der linearen Abbildungen von symmetrischen Tensoren auf symmetrische Tensoren konstruiert werden. Die 36 Komponenten der Tensoren vierter Stufe aus
können als Voigt'scher Notation in eine 6×6-Matrix einsortiert werden:
![{\displaystyle {\stackrel {4}{\mathbf {A} }}=A_{uv}\mathbf {S} _{u}\otimes \mathbf {S} _{v}{\hat {=}}{\begin{bmatrix}A_{11}&A_{12}&A_{13}&A_{14}&A_{15}&A_{16}\\A_{21}&A_{22}&A_{23}&A_{24}&A_{25}&A_{26}\\A_{31}&A_{32}&A_{33}&A_{34}&A_{35}&A_{36}\\A_{41}&A_{42}&A_{43}&A_{44}&A_{45}&A_{46}\\A_{51}&A_{52}&A_{53}&A_{54}&A_{55}&A_{56}\\A_{61}&A_{62}&A_{63}&A_{64}&A_{65}&A_{66}\end{bmatrix}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80oqa4ygw1oqoOoAoQoNrBa2hFa2sNntFCntCQajo2z2wNzNs4aqnB)
Die Vektoren und Matrizen in Voigt'scher Notation können addiert, subtrahiert und mit einem Skalar multipliziert werden. Beim Matrizenprodukt in Voigt'scher Notation muss eine Diagonalmatrix
![{\displaystyle I=\mathrm {diag} (1,1,1,2,2,2)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AnAaNz2e4ygo3aNBBajKNzDnDzDsNnti5a2dCzjaPzjlDzge4zAiQ)
mit den Einträgen
zwischengeschaltet werden:
![{\displaystyle \mathbf {A} :\mathbf {B} =[\mathbf {A} ]^{\top }I[\mathbf {B} ]=A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}+2A_{4}B_{4}+2A_{5}B_{5}+2A_{6}B_{6}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PajeNoDFCygiPatw0ngw0zthDzDzFzDw3nDlCyjCOntG2ajvFatsP)
![{\displaystyle \left[{\stackrel {4}{\mathbf {A} }}:\mathbf {T} \right]=\left[{\stackrel {4}{\mathbf {A} }}\right]I[\mathbf {T} ]}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FzgzFzqw1atdFnqzAnqw4yta4zDoNaDsPnti1ytm4zDaOzDa3njG3)
![{\displaystyle \left[{\stackrel {4}{\mathbf {A} }}:{\stackrel {4}{\mathbf {B} }}\right]=\left[{\stackrel {4}{\mathbf {A} }}\right]I\left[{\stackrel {4}{\mathbf {B} }}\right]}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnjrAzjaNntFBage1nqe1zNs4o2wNyqo0otK0njJFoAzAaDi5yjhB)
Darin steht [x] für die Voigt-Notation von x.
- ↑ P. B. Denton, S. J. Parke, T. Tao, X. Zhang: Eigenvectors from Eigenvalues. (PDF) 10. August 2019, S. 1–3, abgerufen am 29. November 2019 (englisch).
- ↑ J. Hanson: Rotations in three, four, and five dimensions. Bei: arxiv.org. S. 4f.
- Holm Altenbach: Kontinuumsmechanik. Einführung in die materialunabhängigen und materialabhängigen Gleichungen. 2. Auflage. Springer Vieweg, Berlin u. a. 2012, ISBN 978-3-642-24118-5.
- Philippe Ciarlet: Mathematical Elasticity. Band 1: Three-Dimensional Elasticity. North-Holland, Amsterdam 1988, ISBN 0-444-70259-8.
- Wolfgang Ehlers: Ergänzung zu den Vorlesungen Technische Mechanik und Höhere Mechanik. Vektor- und Tensorrechnung, Eine Einführung. 2015 (uni-stuttgart.de [PDF; abgerufen am 3. September 2020]).
- Ralf Greve: Kontinuumsmechanik. Ein Grundkurs für Ingenieure und Physiker. Springer, Berlin u. a. 2003, ISBN 3-540-00760-1.