Multilinear map · Lablypedia

$f : V_{1} \times \dots \times V_{n} \to W,$

where $V_{1}, \dots, V_{n}$ ( $n \in Z_{\geq 0}$ ) 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 square matrix is a multilinear function of the columns (or rows); it is also an alternating function of the columns (or rows).
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 $D^{k} F : R^{m} \times \dots \times R^{m} \to R^{n}$ .

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Coordinate representation

Let

$f : V_{1} \times \dots \times V_{n} \to W,$

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 ${e_{i 1}, \dots, e_{i d_{i}}}$ for each $V_{i}$ and a basis ${b_{1}, \dots, b_{d}}$ for $W$ (using bold for vectors), then we can define a collection of scalars $A_{j_{1} \dots j_{n}}^{k}$ by

$f (e_{1 j_{1}}, \dots, e_{n j_{n}}) = A_{j_{1} \dots j_{n}}^{1} b_{1} + \dots + A_{j_{1} \dots j_{n}}^{d} b_{d} .$

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

$v_{i} = \sum_{j = 1}^{d_{i}} v_{i j} e_{i j}$

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

$f (v_{1}, \dots, v_{n}) = \sum_{j_{1} = 1}^{d_{1}} \dots \sum_{j_{n} = 1}^{d_{n}} \sum_{k = 1}^{d} A_{j_{1} \dots j_{n}}^{k} v_{1 j_{1}} \dots v_{n j_{n}} b_{k} .$

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 ${e_{i 1}, \dots, e_{i d_{i}}} = {e_{1}, e_{2}} = {(1, 0), (0, 1)} .$ Let

$g (e_{1 i}, e_{2 j}, e_{3 k}) = f (e_{i}, e_{j}, e_{k}) = A_{i j k},$ 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: ${e_{1}, e_{1}, e_{1}}, {e_{1}, e_{1}, e_{2}}, {e_{1}, e_{2}, e_{1}}, {e_{1}, e_{2}, e_{2}}, {e_{2}, e_{1}, e_{1}}, {e_{2}, e_{1}, e_{2}}, {e_{2}, e_{2}, e_{1}}, {e_{2}, e_{2}, e_{2}} .$

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

$v_{i} = \sum_{j = 1}^{2} v_{i j} e_{i j} = v_{i 1} \times e_{1} + v_{i 2} \times e_{2} = v_{i 1} \times (1, 0) + v_{i 2} \times (0, 1) .$

The function value at an arbitrary collection of three vectors $v_{i} \in R^{2}$ can be expressed as $g (v_{1}, v_{2}, v_{3}) = \sum_{i = 1}^{2} \sum_{j = 1}^{2} \sum_{k = 1}^{2} A_{i j k} v_{1 i} v_{2 j} v_{3 k},$ or in expanded form as $\begin{matrix} g ((a, b), (c, d) & , (e, f)) = a c e \times g (e_{1}, e_{1}, e_{1}) + a c f \times g (e_{1}, e_{1}, e_{2}) \\ + a d e \times g (e_{1}, e_{2}, e_{1}) + a d f \times g (e_{1}, e_{2}, e_{2}) + b c e \times g (e_{2}, e_{1}, e_{1}) + b c f \times g (e_{2}, e_{1}, e_{2}) \\ + b d e \times g (e_{2}, e_{2}, e_{1}) + b d f \times g (e_{2}, e_{2}, e_{2}) . \end{matrix}$

Relation to tensor products

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

$f : V_{1} \times \dots \times V_{n} \to W,$

and linear maps

$F : V_{1} \otimes \dots \otimes V_{n} \to W,$

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

$f (v_{1}, \dots, v_{n}) = F (v_{1} \otimes \dots \otimes v_{n}) .$

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

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

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,

$D (A) = \sum_{1 \leq k_{1} \leq n} \dots \sum_{1 \leq k_{i} \leq n} \dots \sum_{1 \leq k_{n} \leq n} A (1, k_{1}) A (2, k_{2}) \dots A (n, k_{n}) D ({\hat{e}}_{k_{1}}, \dots, {\hat{e}}_{k_{n}}) .$

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.

Examples

Coordinate representation

Example

Relation to tensor products

Multilinear functions on n×n matrices

Example

Properties

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