Vector-valued function of multiple vectors, linear in each argument
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
![{\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FzNCPytC3ntm4ntG4aghCz2hBoDC4ajFCzNhEzts5oAsPzNdBnqe0)
where
(
) and
are vector spaces (or modules over a commutative ring), with the following property: for each
, if all of the variables but
are held constant, then
is a linear function of
.[1] One way to visualize this is to imagine two orthogonal vectors; if one of these vectors is scaled by a factor of 2 while the other remains unchanged, the cross product likewise scales by a factor of two. If both are scaled by a factor of 2, the cross product scales by a factor of
.
A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer
, a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.
If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating k-linear maps. The latter two coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.
- Any bilinear map is a multilinear map. For example, any inner product on a
-vector space is a multilinear map, as is the cross product of vectors in
.
- The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
- If
is a Ck function, then the
th derivative of
at each point
in its domain can be viewed as a symmetric
-linear function
.[citation needed]
Coordinate representation
[edit]
Let
![{\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FzNCPytC3ntm4ntG4aghCz2hBoDC4ajFCzNhEzts5oAsPzNdBnqe0)
be a multilinear map between finite-dimensional vector spaces, where
has dimension
, and
has dimension
. If we choose a basis
for each
and a basis
for
(using bold for vectors), then we can define a collection of scalars
by
![{\displaystyle f({\textbf {e}}_{1j_{1}},\ldots ,{\textbf {e}}_{nj_{n}})=A_{j_{1}\cdots j_{n}}^{1}\,{\textbf {b}}_{1}+\cdots +A_{j_{1}\cdots j_{n}}^{d}\,{\textbf {b}}_{d}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81njm3zDCQnDC1yqa3njm4aNG2ztnCytm5nArEzqrCaDdEnqnCytzE)
Then the scalars
completely determine the multilinear function
. In particular, if
![{\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{d_{i}}v_{ij}{\textbf {e}}_{ij}\!}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ztoPagnAzgs0aDmNajdFajzEyqeQnjJCotFByjK2a2hCzNdCyqdB)
for
, then
![{\displaystyle f({\textbf {v}}_{1},\ldots ,{\textbf {v}}_{n})=\sum _{j_{1}=1}^{d_{1}}\cdots \sum _{j_{n}=1}^{d_{n}}\sum _{k=1}^{d}A_{j_{1}\cdots j_{n}}^{k}v_{1j_{1}}\cdots v_{nj_{n}}{\textbf {b}}_{k}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Poqi1oAzEaNlBoAw2ygs3ztFAnjoPoArEnDrFnAa3yjm1zDKQoqs3)
Let's take a trilinear function
![{\displaystyle g\colon R^{2}\times R^{2}\times R^{2}\to R,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EytwPngw2ztlEoNw1nDo4nqvDzDePato3aDG2oDJFaAvBnjlAzjwP)
where Vi = R2, di = 2, i = 1,2,3, and W = R, d = 1.
A basis for each Vi is
Let
![{\displaystyle g({\textbf {e}}_{1i},{\textbf {e}}_{2j},{\textbf {e}}_{3k})=f({\textbf {e}}_{i},{\textbf {e}}_{j},{\textbf {e}}_{k})=A_{ijk},}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80oAoPnjiOzNdEoNJFoDnDzNhFajwPyjG4nDa1yqs0zNaPyqnDzto2)
where
. In other words, the constant
is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three
), namely:
![{\displaystyle \{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}\}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ntC4ajaOaqa2a2dDzjBFntCPo2nBnqs5ntBCoNnFzDzBztK0nqrD)
Each vector
can be expressed as a linear combination of the basis vectors
![{\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{2}v_{ij}{\textbf {e}}_{ij}=v_{i1}\times {\textbf {e}}_{1}+v_{i2}\times {\textbf {e}}_{2}=v_{i1}\times (1,0)+v_{i2}\times (0,1).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PytrAzNnFothDa2s4aDmNaqhEaqvFoNa0ztsPnjw0otG0oDwOajBF)
The function value at an arbitrary collection of three vectors
can be expressed as
![{\displaystyle g({\textbf {v}}_{1},{\textbf {v}}_{2},{\textbf {v}}_{3})=\sum _{i=1}^{2}\sum _{j=1}^{2}\sum _{k=1}^{2}A_{ijk}v_{1i}v_{2j}v_{3k},}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QagaNoNKOzAdFaDePzqw4njC1nDlFoDoOyto1a2s2zjo0ntdAoNs0)
or in expanded form as
![{\displaystyle {\begin{aligned}g((a,b),(c,d)&,(e,f))=ace\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1})+acf\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+ade\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1})+adf\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2})+bce\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1})+bcf\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+bde\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1})+bdf\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}).\end{aligned}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ajm0oqi3oNw3ate5yjs5zNw4ytCNaDaPagw0zAa4atvCnDK1ytdB)
Relation to tensor products
[edit]
There is a natural one-to-one correspondence between multilinear maps
![{\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FzNCPytC3ntm4ntG4aghCz2hBoDC4ajFCzNhEzts5oAsPzNdBnqe0)
and linear maps
![{\displaystyle F\colon V_{1}\otimes \cdots \otimes V_{n}\to W{\text{,}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PntwOytC1nAoQytoQo2w4nte4zDsOnDs5aDa2aDrDaDwNajhBags2)
where
denotes the tensor product of
. The relation between the functions
and
is given by the formula
![{\displaystyle f(v_{1},\ldots ,v_{n})=F(v_{1}\otimes \cdots \otimes v_{n}).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AoqdCzto5zgs1zNs2ygvFzDlCata5zAhBoqdDaNJEnAs1atG2zNBC)
Multilinear functions on n×n matrices
[edit]
One can consider multilinear functions, on an n×n matrix over a commutative ring K with identity, as a function of the rows (or equivalently the columns) of the matrix. Let A be such a matrix and ai, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as
![{\displaystyle D(A)=D(a_{1},\ldots ,a_{n}),}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OnjC1nAoNzts3ztBCytm1zDJDytvEoqe1aAwOoAs5agePzNm0zgeN)
satisfying
![{\displaystyle D(a_{1},\ldots ,ca_{i}+a_{i}',\ldots ,a_{n})=cD(a_{1},\ldots ,a_{i},\ldots ,a_{n})+D(a_{1},\ldots ,a_{i}',\ldots ,a_{n}).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OoAiNoAo4ntK2zDzBnDhDnDo2njCOaNo0ntFDoDvAajs3yjwOoAhF)
If we let
represent the jth row of the identity matrix, we can express each row ai as the sum
![{\displaystyle a_{i}=\sum _{j=1}^{n}A(i,j){\hat {e}}_{j}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FzjaOoDa1oDm5zje5yjzCzNnCyqa5z2hBzAoPnqeOoDa5ytK5oDzF)
Using the multilinearity of D we rewrite D(A) as
![{\displaystyle D(A)=D\left(\sum _{j=1}^{n}A(1,j){\hat {e}}_{j},a_{2},\ldots ,a_{n}\right)=\sum _{j=1}^{n}A(1,j)D({\hat {e}}_{j},a_{2},\ldots ,a_{n}).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CyqrCoArAnDa5ajBEngi3yts4ngo0ato4zgwNajC5a2dAnDnBo2wO)
Continuing this substitution for each ai we get, for 1 ≤ i ≤ n,
![{\displaystyle D(A)=\sum _{1\leq k_{1}\leq n}\ldots \sum _{1\leq k_{i}\leq n}\ldots \sum _{1\leq k_{n}\leq n}A(1,k_{1})A(2,k_{2})\dots A(n,k_{n})D({\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}).}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DaghCntK5atG4zNi0aDCNotlDaNhFoqnBatBDngsNnji1ajlEnteQ)
Therefore, D(A) is uniquely determined by how D operates on
.
In the case of 2×2 matrices, we get
![{\displaystyle D(A)=A_{1,1}A_{1,2}D({\hat {e}}_{1},{\hat {e}}_{1})+A_{1,1}A_{2,2}D({\hat {e}}_{1},{\hat {e}}_{2})+A_{1,2}A_{2,1}D({\hat {e}}_{2},{\hat {e}}_{1})+A_{1,2}A_{2,2}D({\hat {e}}_{2},{\hat {e}}_{2}),\,}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82ntnBoAwNzNKOz2aQngi3njzFatC5zta3nDo0ygdDzgi1njKPaAi1)
where
and
. If we restrict
to be an alternating function, then
and
. Letting
, we get the determinant function on 2×2 matrices:
![{\displaystyle D(A)=A_{1,1}A_{2,2}-A_{1,2}A_{2,1}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83a2w1oqzFnAo4nDo0ntiQa2vBathFnAzCzjrEajiNzta1aqhBzqzB)
- A multilinear map has a value of zero whenever one of its arguments is zero.