Multilinear map

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

$f:V_1\times \cdots \times V_n\to W\text{,}$

where $V_1,\dots ,V_n$ ($n\in {\mathbb{Z}}_{\ge 0}$) and $W$ are vector spaces (or modules over a commutative ring), with the following property: for each $i$, if all of the variables but $v_i$ are held constant, then $f(v_1,\dots ,v_i,\dots ,v_n)$ is a linear function of $v_i$. 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 $2^2$.

A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer $k$, 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 $ℝ$-vector space is a multilinear map, as is the cross product of vectors in ${ℝ}^3$.
• The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
• If $F:{ℝ}^m\to {ℝ}^n$ is a C^k function, then the $k$th derivative of $F$ at each point $p$ in its domain can be viewed as a symmetric $k$-linear function $D^{k}F:{ℝ}^m\times \cdots \times {ℝ}^m\to {ℝ}^n$.Template:Citation needed

Coordinate representation

Let

$f:V_1\times \cdots \times V_n\to W\text{,}$

be a multilinear map between finite-dimensional vector spaces, where $V_i$ has dimension $d_i$, and $W$ has dimension $d$. If we choose a basis $\{{\mathbf{\text{e}}}_{i1},\dots ,{\mathbf{\text{e}}}_{i{d}_i}\}$ for each $V_i$ and a basis $\{{\mathbf{\text{b}}}_1,\dots ,{\mathbf{\text{b}}}_d\}$ for $W$ (using bold for vectors), then we can define a collection of scalars $A_{j_1\cdots j_n}^k$ by

$f({\mathbf{\text{e}}}_{1j_1},\dots ,{\mathbf{\text{e}}}_{n{j}_n})=A_{j_1\cdots j_n}^1{\mathbf{\text{b}}}_1+\cdots +A_{j_1\cdots j_n}^d{\mathbf{\text{b}}}_d.$

Then the scalars $\{A_{j_1\cdots j_n}^k\mid 1\le j_i\le d_i,1\le k\le d\}$ completely determine the multilinear function $f$. In particular, if

${\mathbf{\text{v}}}_i=\sum _{j=1}^{d_i}{v}_{ij}{\mathbf{\text{e}}}_{ij}$

for $1\le i\le n$, then

$f({\mathbf{\text{v}}}_1,\dots ,{\mathbf{\text{v}}}_n)=\sum _{j_1=1}^{d_1}\cdots \sum _{j_n=1}^{d_n}\sum _{k=1}^d{A}_{j_1\cdots j_n}^k{v}_{1j_1}\cdots v_{n{j}_n}{\mathbf{\text{b}}}_k.$

Example

Let's take a trilinear function

$g:R^2\times R^2\times R^2\to R,$

where Template:Math, and Template:Math.

A basis for each Template:Mvar is $\{{\mathbf{\text{e}}}_{i1},\dots ,{\mathbf{\text{e}}}_{i{d}_i}\}=\{{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_2\}=\{(1,0),(0,1)\}.$ Let

$g({\mathbf{\text{e}}}_{1i},{\mathbf{\text{e}}}_{2j},{\mathbf{\text{e}}}_{3k})=f({\mathbf{\text{e}}}_i,{\mathbf{\text{e}}}_j,{\mathbf{\text{e}}}_k)=A_{ijk},$

where $i,j,k\in \{1,2\}$. In other words, the constant $A_{ijk}$ is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three $V_i$), namely:

$\{{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_1\},\{{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_2\},\{{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_1\},\{{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_2\},\{{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_1\},\{{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_2\},\{{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_1\},\{{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_2\}.$

Each vector ${\mathbf{\text{v}}}_i\in V_i=R^2$ can be expressed as a linear combination of the basis vectors

${\mathbf{\text{v}}}_i=\sum _{j=1}^2{v}_{ij}{\mathbf{\text{e}}}_{ij}=v_{i1}\times {\mathbf{\text{e}}}_1+v_{i2}\times {\mathbf{\text{e}}}_2=v_{i1}\times (1,0)+v_{i2}\times (0,1).$

The function value at an arbitrary collection of three vectors ${\mathbf{\text{v}}}_i\in R^2$ can be expressed as

$g({\mathbf{\text{v}}}_1,{\mathbf{\text{v}}}_2,{\mathbf{\text{v}}}_3)=\sum _{i=1}^2\sum _{j=1}^2\sum _{k=1}^2{A}_{ijk}{v}_{1i}{v}_{2j}{v}_{3k},$

or in expanded form as

$\begin{array}{cc}g((a,b),(c,d)& ,(e,f))=ace\times g({\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_1)+acf\times g({\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_2)\\ & +ade\times g({\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_1)+adf\times g({\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_2)+bce\times g({\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_1)+bcf\times g({\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_1,{\mathbf{\text{e}}}_2)\\ & +bde\times g({\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_1)+bdf\times g({\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_2,{\mathbf{\text{e}}}_2).\end{array}$

Relation to tensor products

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

$f:V_1\times \cdots \times V_n\to W\text{,}$

and linear maps

$F:V_1\otimes \cdots \otimes V_n\to W\text{,}$

where $V_1\otimes \cdots \otimes V_n$ denotes the tensor product of $V_1,\dots ,V_n$. The relation between the functions $f$ and $F$ is given by the formula

$f(v_1,\dots ,v_n)=F(v_1\otimes \cdots \otimes v_n).$

Multilinear functions on n×n matrices

One can consider multilinear functions, on an Template:Math matrix over a commutative ring Template:Mvar with identity, as a function of the rows (or equivalently the columns) of the matrix. Let Template:Math be such a matrix and Template:Math, be the rows of Template:Math. Then the multilinear function Template:Math can be written as

$D(A)=D(a_1,\dots ,a_n),$

satisfying

$D(a_1,\dots ,c{a}_i+a_{i^{\prime }},\dots ,a_n)=cD(a_1,\dots ,a_i,\dots ,a_n)+D(a_1,\dots ,a_{i^{\prime }},\dots ,a_n).$

If we let ${\hat{e}}_j$ represent the Template:Mvarth row of the identity matrix, we can express each row Template:Math as the sum

$a_i=\sum _{j=1}^{n}A(i,j){\hat{e}}_j.$

Using the multilinearity of Template:Math we rewrite Template:Math as

$D(A)=D(\sum _{j=1}^{n}A(1,j){\hat{e}}_j,a_2,\dots ,a_n)=\sum _{j=1}^{n}A(1,j)D({\hat{e}}_j,a_2,\dots ,a_n).$

Continuing this substitution for each Template:Math we get, for Template:Math,

$D(A)=\sum _{1\le k_1\le n}\dots \sum _{1\le k_i\le n}\dots \sum _{1\le k_n\le n}A(1,k_1)A(2,k_2)\dots A(n,k_n)D({\hat{e}}_{k_1},\dots ,{\hat{e}}_{k_n}).$

Therefore, Template:Math is uniquely determined by how Template:Mvar operates on ${\hat{e}}_{k_1},\dots ,{\hat{e}}_{k_n}$.

Example

In the case of 2×2 matrices, we get

$D(A)=A_{1,1}{A}_{1,2}D({\hat{e}}_1,{\hat{e}}_1)+A_{1,1}{A}_{2,2}D({\hat{e}}_1,{\hat{e}}_2)+A_{1,2}{A}_{2,1}D({\hat{e}}_2,{\hat{e}}_1)+A_{1,2}{A}_{2,2}D({\hat{e}}_2,{\hat{e}}_2),$

where ${\hat{e}}_1=[1,0]$ and ${\hat{e}}_2=[0,1]$. If we restrict $D$ to be an alternating function, then $D({\hat{e}}_1,{\hat{e}}_1)=D({\hat{e}}_2,{\hat{e}}_2)=0$ and $D({\hat{e}}_2,{\hat{e}}_1)=-D({\hat{e}}_1,{\hat{e}}_2)=-D(I)$. Letting $D(I)=1$, we get the determinant function on 2×2 matrices:

$D(A)=A_{1,1}{A}_{2,2}-A_{1,2}{A}_{2,1}.$

Properties

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

See also

• Algebraic form
• Multilinear form
• Homogeneous polynomial
• Homogeneous function
• Tensors