# Multilinear map

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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

${\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}$

where ${\displaystyle V_{1},\ldots ,V_{n}}$ and ${\displaystyle W\!}$ are vector spaces (or modules), with the following property: for each ${\displaystyle i\!}$, if all of the variables but ${\displaystyle v_{i}\!}$ are held constant, then ${\displaystyle f(v_{1},\ldots ,v_{n})}$ is a linear function of ${\displaystyle v_{i}\!}$.[1]

A multilinear map of two variables is a bilinear map. More generally, 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 coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.

## Coordinate representation

Let

${\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}$

be a multilinear map between finite-dimensional vector spaces, where ${\displaystyle V_{i}\!}$ has dimension ${\displaystyle d_{i}\!}$, and ${\displaystyle W\!}$ has dimension ${\displaystyle d\!}$. If we choose a basis ${\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}}$ for each ${\displaystyle V_{i}\!}$ and a basis ${\displaystyle \{{\textbf {b}}_{1},\ldots ,{\textbf {b}}_{d}\}}$ for ${\displaystyle W\!}$ (using bold for vectors), then we can define a collection of scalars ${\displaystyle A_{j_{1}\cdots j_{n}}^{k}}$ by

${\displaystyle f({\textbf {e}}_{1j_{1}},\ldots ,{\textbf {e}}_{nj_{n}})=A_{j_{1}\cdots j_{n}}^{1}\,{\textbf {b}}_{1}+\cdots +A_{j_{1}\cdots j_{n}}^{d}\,{\textbf {b}}_{d}.}$

Then the scalars ${\displaystyle \{A_{j_{1}\cdots j_{n}}^{k}\mid 1\leq j_{i}\leq d_{i},1\leq k\leq d\}}$ completely determine the multilinear function ${\displaystyle f\!}$. In particular, if

${\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{d_{i}}v_{ij}{\textbf {e}}_{ij}\!}$
${\displaystyle f({\textbf {v}}_{1},\ldots ,{\textbf {v}}_{n})=\sum _{j_{1}=1}^{d_{1}}\cdots \sum _{j_{n}=1}^{d_{n}}\sum _{k=1}^{d}A_{j_{1}\cdots j_{n}}^{k}v_{1j_{1}}\cdots v_{nj_{n}}{\textbf {b}}_{k}.}$

## Example

Let's take a trilinear function:

${\displaystyle f\colon R^{2}\times R^{2}\times R^{2}\to R}$
${\displaystyle f({\textbf {e}}_{1i},{\textbf {e}}_{2j},{\textbf {e}}_{3k})=f({\textbf {e}}_{i},{\textbf {e}}_{j},{\textbf {e}}_{k})=A_{ijk}}$, where ${\displaystyle i,j,k\in \{1,2\}}$. In other words, the constant ${\displaystyle A_{ijk}}$ means a function value at one of 8 possible combinations of basis vectors, one per each ${\displaystyle V_{i}}$:

Each vector ${\displaystyle {\textbf {v}}_{i}\in V_{i}=R^{2}}$ can be expressed as a linear combination of the basis vectors:

${\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{2}v_{ij}{\textbf {e}}_{ij}=v_{i1}\times {\textbf {e}}_{1}+v_{i2}\times {\textbf {e}}_{2}=v_{i1}\times (1,0)+v_{i2}\times (0,1)\!}$

The function value at an arbitrary collection of 3 vectors ${\displaystyle {\textbf {v}}_{i}\in R^{2}}$ can be expressed:

${\displaystyle f({\textbf {v}}_{1},{\textbf {v}}_{2},{\textbf {v}}_{3})=\sum _{i=1}^{2}\sum _{j=1}^{2}\sum _{k=1}^{2}A_{ijk}v_{1i}v_{2j}v_{3k}}$.
${\displaystyle f((a,b),(c,d),(e,f))=ace\times f({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1})+acf\times f({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2})+ade\times f({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1})+adf\times f({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2})+bce\times f({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1})+bcf\times f({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2})+bde\times f({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1})+bdf\times f({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2})}$.

## Relation to tensor products

There is a natural one-to-one correspondence between multilinear maps

${\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}$

and linear maps

${\displaystyle F\colon V_{1}\otimes \cdots \otimes V_{n}\to W{\text{,}}}$

where ${\displaystyle V_{1}\otimes \cdots \otimes V_{n}\!}$ denotes the tensor product of ${\displaystyle V_{1},\ldots ,V_{n}}$. The relation between the functions ${\displaystyle f\!}$ and ${\displaystyle F\!}$ is given by the formula

${\displaystyle F(v_{1}\otimes \cdots \otimes v_{n})=f(v_{1},\ldots ,v_{n}).}$

## 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 ${\displaystyle a_{i}}$, 1 ≤ in be the rows of A. Then the multilinear function D can be written as

${\displaystyle D(A)=D(a_{1},\ldots ,a_{n})\,}$

satisfying

${\displaystyle D(a_{1},\ldots ,ca_{i}+a_{i}',\ldots ,a_{n})=cD(a_{1},\ldots ,a_{i},\ldots ,a_{n})+D(a_{1},\ldots ,a_{i}',\ldots ,a_{n})\,}$

If we let ${\displaystyle {\hat {e}}_{j}}$ represent the jth row of the identity matrix we can express each row ${\displaystyle a_{i}}$ as the sum

${\displaystyle a_{i}=\sum _{j=1}^{n}A(i,j){\hat {e}}_{j}}$

Using the multilinearity of D we rewrite D(A) as

${\displaystyle D(A)=D\left(\sum _{j=1}^{n}A(1,j){\hat {e}}_{j},a_{2},\ldots ,a_{n}\right)=\sum _{j=1}^{n}A(1,j)D({\hat {e}}_{j},a_{2},\ldots ,a_{n})}$

Continuing this substitution for each ${\displaystyle a_{i}}$ we get, for 1 ≤ in

${\displaystyle D(A)=\sum _{1\leq k_{i}\leq n}A(1,k_{1})A(2,k_{2})\dots A(n,k_{n})D({\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}})}$
where, since in our case ${\displaystyle 1\leq i\leq n}$
${\displaystyle \sum _{1\leq k_{i}\leq n}=\sum _{1\leq k_{1}\leq n}\ldots \sum _{1\leq k_{i}\leq n}\ldots \sum _{1\leq k_{n}\leq n}\,}$
as a series of nested summations.

Therefore, D(A) is uniquely determined by how ${\displaystyle D}$ operates on ${\displaystyle {\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}}$.

## Example

In the case of 2×2 matrices we get

${\displaystyle D(A)=A_{1,1}A_{2,1}D({\hat {e}}_{1},{\hat {e}}_{1})+A_{1,1}A_{2,2}D({\hat {e}}_{1},{\hat {e}}_{2})+A_{1,2}A_{2,1}D({\hat {e}}_{2},{\hat {e}}_{1})+A_{1,2}A_{2,2}D({\hat {e}}_{2},{\hat {e}}_{2})\,}$

Where ${\displaystyle {\hat {e}}_{1}=[1,0]}$ and ${\displaystyle {\hat {e}}_{2}=[0,1]}$. If we restrict D to be an alternating function then ${\displaystyle D({\hat {e}}_{1},{\hat {e}}_{1})=D({\hat {e}}_{2},{\hat {e}}_{2})=0}$ and ${\displaystyle D({\hat {e}}_{2},{\hat {e}}_{1})=-D({\hat {e}}_{1},{\hat {e}}_{2})=-D(I)}$. Letting ${\displaystyle D(I)=1}$ we get the determinant function on 2×2 matrices:

${\displaystyle D(A)=A_{1,1}A_{2,2}-A_{1,2}A_{2,1}\,}$

## Properties

A multilinear map has a value of zero whenever one of its arguments is zero.

For n>1, the only n-linear map which is also a linear map is the zero function, see bilinear map#Examples.

## References

1. Lang. Algebra. Springer; 3rd edition (January 8, 2002)