Multilinear map

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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

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.

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: