Content deleted Content added
Undid revision 1077745741 by 2601:19B:700:CEF0:31A2:C7E5:2CC1:16FB (talk) the removed paragraph says nothing about eigenvalues of A itself |
fixed numerous minor math typesetting errors |
||
| (35 intermediate revisions by 21 users not shown) | |||
Line 1:
{{Short description|Result about when a matrix can be diagonalized}}
In [[mathematics]], particularly [[linear algebra]] and [[functional analysis]], a '''spectral theorem''' is a result about when a [[linear operator]] or [[matrix (mathematics)|matrix]] can be [[Diagonalizable matrix|diagonalized]] (that is, represented as a [[diagonal matrix]] in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on [[finite-dimensional vector
Examples of operators to which the spectral theorem applies are [[self-adjoint operator]]s or more generally [[normal operator]]s on [[Hilbert space]]s.
Line 6:
The spectral theorem also provides a [[canonical form|canonical]] decomposition, called the '''[[eigendecomposition of a matrix|spectral decomposition]]''', of the underlying vector space on which the operator acts.
[[Augustin-Louis Cauchy]] proved the spectral theorem for [[Symmetric matrix|symmetric matrices]], i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systematic about
This article mainly focuses on the simplest kind of spectral theorem, that for a [[self-adjoint]] operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.
== Finite-dimensional case ==
<!-- This section is linked from [[Singular value decomposition]] --> === Hermitian maps and Hermitian matrices ===
We begin by considering a [[Hermitian matrix]] on <math>\mathbb{C}^n</math> (but the following discussion will be adaptable to the more restrictive case of [[symmetric matrix|symmetric matrices]] on
An equivalent condition is that {{math| ''A''{{sup|*}} {{=}} ''A'' }}, where {{math| ''A''{{sup|*}} }} is the [[Hermitian conjugate]] of {{math|''A''}}. In the case that {{math|''A''}} is identified with a Hermitian matrix, the matrix of {{math| ''A''{{sup|*}} }} is equal to its [[conjugate transpose]]. (If {{math|''A''}} is a [[real matrix]], then this is equivalent to {{math| ''A''{{sup|T}} {{=}} ''A''}}, that is, {{math|''A''}} is a [[symmetric matrix]].)
▲:<math> \langle A x, y \rangle = \langle x, A y \rangle.</math>
▲'''Theorem'''. If {{math|''A''}} is Hermitian, then there exists an [[orthonormal basis]] of {{math|''V''}} consisting of eigenvectors of {{math|''A''}}. Each eigenvalue is real.
We provide a sketch of a proof for the case where the underlying field of scalars is the [[complex number]]s.
By the [[fundamental theorem of algebra]], applied to the [[characteristic polynomial]] of {{math|''A''}}, there is at least one complex eigenvalue {{math| ''λ''
we find that {{math| ''λ''
The matrix representation of {{math|''A''}} in a basis of eigenvectors is diagonal, and by the construction the proof gives a basis of mutually orthogonal eigenvectors; by choosing them to be unit vectors one obtains an orthonormal basis of eigenvectors. {{math|''A''}} can be written as a linear combination of pairwise orthogonal projections, called its '''spectral decomposition'''. Let
be the eigenspace corresponding to an eigenvalue
When the matrix being decomposed is Hermitian, the spectral decomposition is a special case of the [[Schur decomposition]] (see the proof in case of [[#Normal matrices|normal matrices]] below).
▲: <math>V_\lambda = \{v \in V: A v = \lambda v\}</math>
▲be the eigenspace corresponding to an eigenvalue {{math|''λ''}}. Note that the definition does not depend on any choice of specific eigenvectors. {{math|''V''}} is the orthogonal direct sum of the spaces {{math|''V''<sub>''λ''</sub>}} where the index ranges over eigenvalues.
The spectral decomposition is a special case of the [[singular value decomposition]], which states that any matrix <math>\ A \in \mathbb{C}^{m \times n}\ </math> can be expressed as
<math>\ A = U\ \Sigma\ V^{*}\ ,</math> where <math>\ U \in \mathbb{C}^{m \times m}\ </math> and <math>\ V \in \mathbb{C}^{n \times n}\ </math> are [[unitary matrices]] and <math>\ \Sigma \in \mathbb{R}^{m \times n}\ </math> is a diagonal matrix. The diagonal entries of <math>\ \Sigma\ </math> are uniquely determined by <math>\ A\ </math> and are known as the [[singular values]] of <math>\ A ~.</math> If <math>\ A\ </math> is Hermitian, then <math>\ A^* = A\ </math> and <math>\ V\ \Sigma\ U^* = U\ \Sigma\ V^*\ </math> which implies <math>\ U = V ~.</math>
=== Normal matrices ===
{{main|Normal matrix}}
The spectral theorem extends to a more general class of matrices. Let {{math|''A''}} be an operator on a finite-dimensional inner product space. {{math|''A''}} is said to be [[normal matrix|normal]] if {{nobr|{{math|
One can show that {{math|''A''}} is normal if and only if it is unitarily diagonalizable If {{math|''A''}} is normal, then one sees that {{nobr|{{math|
In other words, {{math|''A''}} is normal if and only if there exists a [[unitary matrix]] {{math|''U''}} such that
▲: <math>A = U D U^*,</math>
where {{math|''D''}} is a [[diagonal matrix]]. Then, the entries of the diagonal of {{math|''D''}} are the [[eigenvalue]]s of {{math|''A''}}. The column vectors of {{math|''U''}} are the eigenvectors of {{math|''A''}} and they are orthonormal. Unlike the Hermitian case, the entries of {{math|''D''}} need not be real.
== Compact self-adjoint operators ==
{{
In the more general setting of Hilbert spaces, which may have an infinite dimension, the statement of the spectral theorem for [[compact operator|compact]] [[self-adjoint operators]] is virtually the same as in the finite-dimensional case.
As for Hermitian matrices, the key point is to prove the existence of at least one nonzero eigenvector. One cannot rely on determinants to show existence of eigenvalues, but one can use a maximization argument analogous to the variational characterization of eigenvalues.
If the compactness assumption is removed, then it is ''not'' true that every self-adjoint operator has eigenvectors. For example, the multiplication operator <math>M_{x}</math> on <math>L^2([0,1])</math> which takes each <math>\psi(x) \in L^2([0,1])</math> to <math>x\psi(x)</math> is bounded and self-adjoint, but has no eigenvectors. However, its spectrum, suitably defined, is still equal to <math>[0,1]</math>, see [[Spectrum_(functional_analysis)#Spectrum_of_a_bounded_operator| spectrum of bounded operator]].
== Bounded self-adjoint operators ==
Line 73 ⟶ 65:
===Possible absence of eigenvectors===
The next generalization we consider is that of [[
This operator does not have any eigenvectors ''in'' <math>L^2([0,1])</math>, though it does have eigenvectors in a larger space. Namely the [[Distribution (mathematics)|distribution]] <math>\varphi(t)=\delta(t-t_0)</math>, where <math>\delta</math> is the [[Dirac delta function]], is an eigenvector when construed in an appropriate sense. The Dirac delta function is however not a function in the classical sense and does not lie in the Hilbert space {{math|''L''<sup>2</sup>[0, 1]}} or any other [[Banach space]]. Thus, the delta-functions are "generalized eigenvectors" of <math>A</math> but not eigenvectors in the usual sense. ▼
▲:<math> [A \varphi](t) = t \varphi(t). \;</math>
▲This operator does not have any eigenvectors ''in'' <math>L^2([0,1])</math>, though it does have eigenvectors in a larger space. Namely the [[Distribution (mathematics)|distribution]] <math>\varphi(t)=\delta(t-t_0)</math>, where <math>\delta</math> is the [[Dirac delta function]], is an eigenvector when construed in an appropriate sense. The Dirac delta function is however not a function in the classical sense and does not lie in the Hilbert space {{math|''L''<sup>2</sup>[0, 1]}} or any other [[Banach space]]. Thus, the delta-functions are "generalized eigenvectors" of <math>A</math> but not eigenvectors in the usual sense.
===Spectral subspaces and projection-valued measures===
In the absence of (true) eigenvectors, one can look for
One formulation of the spectral theorem expresses the operator {{math|''A''}} as an integral of the coordinate function over the operator's [[Eigenvector#Infinite dimensions|spectrum]] <math>\sigma(A)</math> with respect to a projection-valued measure.<ref>{{harvnb|Hall|2013}} Theorem 7.12</ref>▼
▲One formulation of the spectral theorem expresses the operator {{math|''A''}} as an integral of the coordinate function over the operator's
When the self-adjoint operator in question is [[compact operator|compact]], this version of the spectral theorem reduces to something similar to the finite-dimensional spectral theorem above, except that the operator is expressed as a finite or countably infinite linear combination of projections, that is, the measure consists only of atoms.
Line 93 ⟶ 83:
An alternative formulation of the spectral theorem says that every bounded self-adjoint operator is unitarily equivalent to a multiplication operator. The significance of this result is that multiplication operators are in many ways easy to understand.
{{math theorem|name='''Theorem'''
<math display="block"> U^* T U = A,</math>
where {{math|''T''}} is the [[multiplication operator]]:
Line 99 ⟶ 89:
and <math>\|T\| = \|f\|_\infty</math>.}}
The spectral theorem is the beginning of the vast research area of functional analysis called [[operator theory]]; see also the [[
There is also an analogous spectral theorem for bounded [[normal operator]]s on Hilbert spaces. The only difference in the conclusion is that now {{math|''f''}} may be complex-valued.
Line 113 ⟶ 103:
for some measure <math>\mu</math> and some family <math>\{H_{\lambda}\}</math> of Hilbert spaces. The measure <math>\mu</math> is uniquely determined by <math>A</math> up to measure-theoretic equivalence; that is, any two measure associated to the same <math>A</math> have the same sets of measure zero. The dimensions of the Hilbert spaces <math>H_{\lambda}</math> are uniquely determined by <math>A</math> up to a set of <math>\mu</math>-measure zero.}}
The spaces <math>H_{\lambda}</math> can be thought of as something like "eigenspaces" for <math>A</math>. Note, however, that unless the one-element set <math>
Although both the multiplication-operator and direct integral formulations of the spectral theorem express a self-adjoint operator as unitarily equivalent to a multiplication operator, the direct integral approach is more canonical. First, the set over which the direct integral takes place (the spectrum of the operator) is canonical. Second, the function we are multiplying by is canonical in the direct-integral approach: Simply the function <math>\lambda\mapsto\lambda</math>.
Line 120 ⟶ 110:
A vector <math>\varphi</math> is called a [[cyclic vector]] for <math>A</math> if the vectors <math>\varphi,A\varphi,A^2\varphi,\ldots</math> span a dense subspace of the Hilbert space. Suppose <math>A</math> is a bounded self-adjoint operator for which a cyclic vector exists. In that case, there is no distinction between the direct-integral and multiplication-operator formulations of the spectral theorem. Indeed, in that case, there is a measure <math>\mu</math> on the spectrum <math>\sigma(A)</math> of <math>A</math> such that <math>A</math> is unitarily equivalent to the "multiplication by <math>\lambda</math>" operator on <math>L^2(\sigma(A),\mu)</math>.<ref>{{harvnb|Hall|2013}} Lemma 8.11</ref> This result represents <math>A</math> simultaneously as a multiplication operator ''and'' as a direct integral, since <math>L^2(\sigma(A),\mu)</math> is just a direct integral in which each Hilbert space <math>H_{\lambda}</math> is just <math>\mathbb{C}</math>.
Not every bounded self-adjoint operator admits a cyclic vector; indeed, by the uniqueness in the direct integral decomposition, this can occur only when all the <math>H_{\lambda}</math>'s have dimension one. When this happens, we say that <math>A</math> has "simple spectrum" in the sense of [[Self-
Although not every <math>A</math> admits a cyclic vector,
===Functional calculus===
One important application of the spectral theorem (in whatever form) is the idea of defining a [[functional calculus]]. That is, given a function <math>f</math> defined on the spectrum of <math>A</math>, we wish to define an operator <math>f(A)</math>. If <math>f</math> is simply a positive power, <math>f(x) = x^n</math>, then <math>f(A)</math> is just the <math>n
That is to say, each space <math>H_{\lambda}</math> in the direct integral is a (generalized) eigenspace for <math>f(A)</math> with eigenvalue <math>f(\lambda)</math>.
==
Many important linear operators which occur in [[Mathematical analysis|analysis]], such as [[differential operators]], are [[unbounded operator|unbounded]]. There is also a spectral theorem for [[self-adjoint operator]]s that applies in these cases. To give an example, every constant-coefficient differential operator is unitarily equivalent to a multiplication operator. Indeed, the unitary operator that implements this equivalence is the [[Fourier transform]]; the multiplication operator is a type of [[Multiplier (Fourier analysis)|Fourier multiplier]].
In general, spectral theorem for self-adjoint operators may take several equivalent forms.<ref>See Section 10.1 of {{harvnb|Hall|2013}}</ref> Notably, all of the formulations given in the previous section for bounded self-adjoint operators—the projection-valued measure version, the multiplication-operator version, and the direct-integral version—continue to hold for unbounded self-adjoint operators, with small technical modifications to deal with domain issues. Specifically, the only reason the multiplication operator <math>A</math> on <math>L^2([0,1])</math> is bounded, is due to the choice of domain <math>[0,1]</math>. The same operator on, e.g., <math>L^2(\mathbb{R})</math> would be unbounded. ▼
The notion of "generalized eigenvectors" naturally extends to unbounded self-adjoint operators, as they are characterized as [[Probability_amplitude#Normalization|non-normalizable]] eigenvectors. Contrary to the case of [[Spectral_theorem#Spectral_subspaces_and_projection-valued_measures|almost eigenvectors]], however, the eigenvalues can be real or complex and, even if they are real, do not necessarily belong to the spectrum. Though, for self-adjoint operators there always exist a real subset of "generalized eigenvalues" such that the corresponding set of eigenvectors is [[Total_set|complete]].{{sfn|de la Madrid Modino|2001|pp=95-97}}
▲In general, spectral theorem for self-adjoint operators may take several equivalent forms.<ref>See Section 10.1 of {{harvnb|Hall|2013}}</ref> Notably, all of the formulations given in the previous section for bounded self-adjoint operators—the projection-valued measure version, the multiplication-operator version, and the direct-integral version—continue to hold for unbounded self-adjoint operators, with small technical modifications to deal with domain issues.
== See also ==
Line 150 ⟶ 142:
{{reflist}}
==
{{Reflist}}
* [[Sheldon Axler]], ''Linear Algebra Done Right'', Springer Verlag, 1997
* {{citation | last = Hall |first = B.C. |title = Quantum Theory for Mathematicians|series=Graduate Texts in Mathematics|volume=267 | year = 2013 |publisher = Springer|bibcode = 2013qtm..book.....H |isbn=978-1461471158}}
* [[Paul Halmos]], [https://www.jstor.org/stable/2313117 "What Does the Spectral Theorem Say?"], ''American Mathematical Monthly'', volume 70, number 3 (1963), pages 241–247 [http://www.math.wsu.edu/faculty/watkins/Math502/pdfiles/spectral.pdf Other link]
*{{cite thesis |last=de la Madrid Modino |first= R. |date=2001 |title= Quantum mechanics in rigged Hilbert space language|url=https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en |degree= PhD |publisher= Universidad de Valladolid}}
* [[Michael C. Reed|M. Reed]] and [[Barry Simon|B. Simon]], ''Methods of Mathematical Physics'', vols I–IV, Academic Press 1972.
* [[Gerald Teschl|G. Teschl]], ''Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators'', https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
* {{Cite book |title=Spectral Theory and Quantum Mechanics; Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation 2nd Edition |author= Valter Moretti |author-link= Valter Moretti |publisher= Springer |year=
{{Functional
{{
[[Category:Spectral theory|*]]
| |||