Multilinear

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 .

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 Map s are multilinear maps.

MULTILINEAR FUNCTIONS ON ''N''&TIMES;''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

arepsilon_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)arepsilon_{j}

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

:

D(A) = D\left(\sum_{j=1}^n A(i,j)arepsilon_{j}, a_2, \ldots, a_n
ight)
       = \sum_{j=1}^n A(i,j) D(arepsilon_{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(arepsilon_{k_{1}},\dots,arepsilon_{k_{n}})

So D(A) is uniquely determined by how it operates on

D(arepsilon_{k_{1}},\dots,arepsilon_{k_{n}})

.

In the case of ''2''×''2'' matrices we get

:

D(A) = A_{1,1}A_{2,1}D(arepsilon_1,arepsilon_1) + A_{1,1}A_{2,2}D(arepsilon_1,arepsilon_2) + A_{1,2}A_{2,1}D(arepsilon_2,arepsilon_1) + A_{1,2}A_{2,2}D(arepsilon_2,arepsilon_2) \,

Where

arepsilon_1 = and arepsilon_2 = [0,1

. If we restrict D to be an alternating function then

D(arepsilon_1,arepsilon_1) = D(arepsilon_2,arepsilon_2) = 0

and

D(arepsilon_2,arepsilon_1) = -D(arepsilon_1,arepsilon_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.

For ''n''>1, the only ''n''-linear map which is also a linear map is the Zero Function , see Bilinear Map#Examples .

SEE ALSO

Algebraic Form

Multilinear Form

Homogeneous Polynomial

Homogeneous Function

Tensor s

Information About

Multilinear