Multilinear map

From Wikipedia, the free encyclopedia

This article does not cite any references or sources. (December 2007)
Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed.

In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable.

A multilinear map of n variables is also called an n-linear map.

If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating n-linear maps. The latter coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.

General discussion of where this leads is at multilinear algebra.

1 Examples
2 Multilinear functions on n×n matrices
3 Properties
4 See also

[edit] Examples

An inner product over the real number field (dot product) is a symmetric bilinear function of two vector variables,
The determinant of a matrix is a skew-symmetric multilinear function of the columns (or rows) of a square matrix.
The trace of a matrix is a multilinear function of the columns (or rows) of a square matrix.
Bilinear maps are multilinear maps.

[edit] 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

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

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 $\varepsilon_j$ represent the jth row of the identity matrix we can express each row $a i$ as the sum

$a_{i} = \sum_{j=1}^n A(i,j)\varepsilon_{j}$

Using the multilinearity of D we rewrite D(A) as

$D(A) = D\left(\sum_{j=1}^n A(i,j)\varepsilon_{j}, a_2, \ldots, a_n\right) = \sum_{j=1}^n A(i,j) D(\varepsilon_{j},a_2,\ldots,a_n)$

Continuing this substitution for each $a i$ we get, for 1 ≤ i ≤ n

$D(A) = \sum_{1\le k_i\le n} A(1,k_{1})A(2,k_{2})\dots A(n,k_{n}) D(\varepsilon_{k_{1}},\dots,\varepsilon_{k_{n}})$

So D(A) is uniquely determined by how $D$ operates on $\varepsilon_{k_{1}},\dots,\varepsilon_{k_{n}}$ .

In the case of 2×2 matrices we get

$D(A) = A_{1,1}A_{2,1}D(\varepsilon_1,\varepsilon_1) + A_{1,1}A_{2,2}D(\varepsilon_1,\varepsilon_2) + A_{1,2}A_{2,1}D(\varepsilon_2,\varepsilon_1) + A_{1,2}A_{2,2}D(\varepsilon_2,\varepsilon_2) \,$

Where $\varepsilon_1 = [1,0]$ and $\varepsilon_2 = [0,1]$ . If we restrict D to be an alternating function then $D(\varepsilon_1,\varepsilon_1) = D(\varepsilon_2,\varepsilon_2) = 0$ and $D(\varepsilon_2,\varepsilon_1) = -D(\varepsilon_1,\varepsilon_2) = -D(I)$ . Letting $D (I) = 1$ we get the determinant function on 2×2 matrices: