Multilinear map
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://wikimedia.org/api/rest_v1/media/math/render/svg/a791997e08e880ad7aeb9808d71b559be172eeb4)
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.
Examples
- 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
Let
![{\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a791997e08e880ad7aeb9808d71b559be172eeb4)
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://wikimedia.org/api/rest_v1/media/math/render/svg/5d07690f959c7d0838656d909feb5ed22befd93b)
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://wikimedia.org/api/rest_v1/media/math/render/svg/45610ff4e420302a03b9b0d7da8e8763ad72d9be)
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://wikimedia.org/api/rest_v1/media/math/render/svg/1aa5bcb30ebd68e758fd61bebf5afc7805670ae7)
Example
Let's take a trilinear function
![{\displaystyle g\colon R^{2}\times R^{2}\times R^{2}\to R,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b941dd650bc45f68edc621167286b7a2ded0f441)
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://wikimedia.org/api/rest_v1/media/math/render/svg/4bf1d1272bc7ab6c71a041888f359e47319fc566)
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://wikimedia.org/api/rest_v1/media/math/render/svg/5e980321c63bc49ae91cfeee9e9dc6a63e574eec)
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://wikimedia.org/api/rest_v1/media/math/render/svg/195f76aa1c3e82031ab1dac34551d44a84b4209a)
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://wikimedia.org/api/rest_v1/media/math/render/svg/00c3c726ba2215d8d95f0ab629653e6464e2fc54)
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://wikimedia.org/api/rest_v1/media/math/render/svg/8004aa7c471298597489932310d46c814df7592e)
Relation to tensor products
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://wikimedia.org/api/rest_v1/media/math/render/svg/a791997e08e880ad7aeb9808d71b559be172eeb4)
and linear maps
![{\displaystyle F\colon V_{1}\otimes \cdots \otimes V_{n}\to W{\text{,}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e42995ff0960cd8e28652f5923625c24301e0e6)
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://wikimedia.org/api/rest_v1/media/math/render/svg/fabd5694e57568da60d1396aeabc37bfe518679d)
Multilinear functions on n×n matrices
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://wikimedia.org/api/rest_v1/media/math/render/svg/2d95ff355759d90567c94bab52d2be90b17044b3)
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://wikimedia.org/api/rest_v1/media/math/render/svg/2ba3bf8e7663ef1cf66d92364e8cb4f057842baa)
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://wikimedia.org/api/rest_v1/media/math/render/svg/a432b35b0942983d76d9c97ae6f1eb2b39979b3a)
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://wikimedia.org/api/rest_v1/media/math/render/svg/d9edbeff3909bda7958df41684d30993bff6ecd2)
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://wikimedia.org/api/rest_v1/media/math/render/svg/c0ade79188714293a0c31aafe19cde3d1500be20)
Therefore, D(A) is uniquely determined by how D operates on .
Example
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://wikimedia.org/api/rest_v1/media/math/render/svg/6e6ebd37727c0da7d3a199537f648bc4a5d712a5)
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://wikimedia.org/api/rest_v1/media/math/render/svg/73d5acaff8f64e103de11afcd45b0135351ae5ce)
Properties
- A multilinear map has a value of zero whenever one of its arguments is zero.
See also
References
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