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.

Contents

[edit] Examples

[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 ai, 1 ≤ in 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 ai 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 ai we get, for 1 ≤ in


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:


D(A) = A_{1,1}A_{2,2} - A_{1,2}A_{2,1} \,

[edit] Properties

A multilinear map has a value of zero whenever one of its arguments is zero.

For n>1, the only n-linear map which is also a linear map is the zero function, see bilinear map#Examples.

[edit] See also