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

where $V_{1},\ldots ,V_{n}$ 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 $f(v_{1},\ldots ,v_{n})$ is a linear function of $v_{i}$ .^[1]

A multilinear map of one variable is a linear map, and 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.

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 $\mathbb {R} ^{3}$ .
The determinant of a matrix is an antisymmetric multilinear function of the columns (or rows) of a square matrix.
If $F\colon {\mathbb {R}}^{m}\to {\mathbb {R}}^{n}$ 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 $D^{k}\!f(p)\colon {\mathbb {R}}^{m}\times \cdots \times {\mathbb {R}}^{m}\to {\mathbb {R}}^{n}$ .
The tensor-to-vector projection in multilinear subspace learning is a multilinear map as well.

Coordinate representation

Let

f\colon 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 $\{{\textbf {e}}_{{i1}},\ldots ,{\textbf {e}}_{{id_{i}}}\}$ for each $V_{i}\!$ and a basis $\{{\textbf {b}}_{1},\ldots ,{\textbf {b}}_{d}\}$ for $W\!$ (using bold for vectors), then we can define a collection of scalars $A_{{j_{1}\cdots j_{n}}}^{k}$ by

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

{\textbf {v}}_{i}=\sum _{{j=1}}^{{d_{i}}}v_{{ij}}{\textbf {e}}_{{ij}}\!

for $1\leq i\leq n\!$ , then

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

f\colon R^2 \times R^2 \times R^2 \to R,

where $V i = R 2, d i = 2, i = 1,2,3$ , and $W = R, d = 1$ .

A basis for each $V i$ is $\{\textbf{e}_{i1},\ldots,\textbf{e}_{id_i}\} = \{\textbf{e}_{1}, \textbf{e}_{2}\} = \{(1,0), (0,1)\}.$ Let

f(\textbf{e}_{1i},\textbf{e}_{2j},\textbf{e}_{3k}) = f(\textbf{e}_{i},\textbf{e}_{j},\textbf{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:

\{\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\}.

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

\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 three vectors ${\textbf {v}}_{i}\in R^{2}$ can be expressed as

f(\textbf{v}_1,\textbf{v}_2, \textbf{v}_3) = \sum_{i=1}^{2} \sum_{j=1}^{2} \sum_{k=1}^{2} A_{i j k} v_{1i} v_{2j} v_{3k}.

Or, in expanded form as

\begin{align} 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). \end{align}

Relation to tensor products

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

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

and linear maps

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

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

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 $n \times n$ 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 $a i, 1 \leq i \leq n$ , be the rows of $A$ . Then the multilinear function $D$ can be written as

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

satisfying

D(a_{1},\ldots,c a_{i} + a_{i}',\ldots,a_{n}) = c D(a_{1},\ldots,a_{i},\ldots,a_{n}) + D(a_{1},\ldots,a_{i}',\ldots,a_{n}).

If we let ${\hat {e}}_{j}$ represent the $j$ th 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\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 $a i$ we get, for $1 \leq i \leq n$ ,

D(A) = \sum_{1\le k_i\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}}),

where, since in our case $1 \leq i \leq n$ ,

\sum_{1\le k_i \le n} = \sum_{1\le k_1 \le n} \ldots \sum_{1\le k_i \le n} \ldots \sum_{1\le k_n \le n}

is a series of nested summations.

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_{{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 ${\hat {e}}_{1}=[1,0]$ 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.
A multilinear map has a value of zero whenever its arguments are linearly dependent.

References

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

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Multilinear map

Examples

Coordinate representation

Example

Relation to tensor products

Multilinear functions on n×n matrices

Example

Properties

See also

References