Multilinear map

Short description: Vector-valued function of multiple vectors, linear in each argument

For multilinear maps used in cryptography, see Cryptographic 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 $V_1,\dots ,V_n$ ($n\in {\mathbb{\mathbb{Z}}}_{\ge 0}$) 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 $\mathbb{ℝ}$-vector space is a multilinear map, as is the cross product of vectors in ${\mathbb{ℝ}}^3$.
• The determinant of a square matrix is a multilinear function of the columns (or rows); it is also an alternating function of the columns (or rows).

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 $\{{\text{e}}_{i1},\dots ,{\text{e}}_{i{d}_i}\}$ for each $V_i$ and a basis $\{{\text{b}}_1,\dots ,{\text{b}}_d\}$ for $W$ (using bold for vectors), then we can define a collection of scalars $A_{j_1\cdots j_n}^k$ by

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

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

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

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

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

Example

Let's take a trilinear function

$$g:R^2\times R^2\times R^2\to R,$$ where V_i = R², d_i = 2, i = 1,2,3, and W = R, d = 1.

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

$$g({\text{e}}_{1i},{\text{e}}_{2j},{\text{e}}_{3k})=f({\text{e}}_i,{\text{e}}_j,{\text{e}}_k)=A_{ijk},$$ 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: $$\{{\text{e}}_1,{\text{e}}_1,{\text{e}}_1\},\{{\text{e}}_1,{\text{e}}_1,{\text{e}}_2\},\{{\text{e}}_1,{\text{e}}_2,{\text{e}}_1\},\{{\text{e}}_1,{\text{e}}_2,{\text{e}}_2\},\{{\text{e}}_2,{\text{e}}_1,{\text{e}}_1\},\{{\text{e}}_2,{\text{e}}_1,{\text{e}}_2\},\{{\text{e}}_2,{\text{e}}_2,{\text{e}}_1\},\{{\text{e}}_2,{\text{e}}_2,{\text{e}}_2\}.$$

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

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

The function value at an arbitrary collection of three vectors ${\text{v}}_i\in R^2$ can be expressed as $$g({\text{v}}_1,{\text{v}}_2,{\text{v}}_3)=\sum _{i=1}^2\sum _{j=1}^2\sum _{k=1}^2{A}_{ijk}{v}_{1i}{v}_{2j}{v}_{3k},$$ or in expanded form as $$\begin{array}{rl}g((a,b),(c,d)& ,(e,f))=ace\times g({\text{e}}_1,{\text{e}}_1,{\text{e}}_1)+acf\times g({\text{e}}_1,{\text{e}}_1,{\text{e}}_2)\\ & +ade\times g({\text{e}}_1,{\text{e}}_2,{\text{e}}_1)+adf\times g({\text{e}}_1,{\text{e}}_2,{\text{e}}_2)+bce\times g({\text{e}}_2,{\text{e}}_1,{\text{e}}_1)+bcf\times g({\text{e}}_2,{\text{e}}_1,{\text{e}}_2)\\ & +bde\times g({\text{e}}_2,{\text{e}}_2,{\text{e}}_1)+bdf\times g({\text{e}}_2,{\text{e}}_2,{\text{e}}_2).\end{array}$$

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 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 a_i, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D 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 jth row of the identity matrix, we can express each row a_i as the sum

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

Using the multilinearity of D we rewrite D(A) 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 a_i we get, for 1 ≤ i ≤ n,

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

Therefore, D(A) is uniquely determined by how D 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