Multilinear map

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).
• If F\colon \mathbb{R}^m \to \mathbb{R}^n is a Ck function, then the kth derivative of F at each point p in its domain can be viewed as a symmetric k-linear function D^k!F\colon \mathbb{R}^m\times\cdots\times\mathbb{R}^m \to \mathbb{R}^n.

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

:g\colon R^2 \times R^2 \times R^2 \to R, where , and .

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

:g(\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_{i j k} 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 :g(\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} 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{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,\ldots,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 ai, 1 ≤ i ≤ 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 jth row of the identity matrix, we can express each row ai 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 ai we get, for 1 ≤ i ≤ n,

: D(A) = \sum_{1\le k_1 \le n} \ldots \sum_{1\le k_i \le n} \ldots \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.

References

References

1. Lang, Serge. (2005). "Algebra". Springer.