Multilinear map

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In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function

f\colon V_1 \times \cdots \times V_n \to W\text{,}

where V_1,\ldots,V_n and W\! are vector spaces (or modules), with the following property: for each i\!, if all of the variables but v_i\! are held constant, then f(v_1,\ldots,v_n) is a linear function of v_i\!.[1]

A multilinear map of two variables is a bilinear map. More generally, 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 coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.

Examples[edit]

Coordinate representation[edit]

Let

f\colon V_1 \times \cdots \times V_n \to W\text{,}

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 \{\textbf{e}_{i1},\ldots,\textbf{e}_{id_i}\} for each V_i\! and a basis \{\textbf{b}_1,\ldots,\textbf{b}_d\} for W\! (using bold for vectors), then we can define a collection of scalars A_{j_1\cdots j_n}^k by

f(\textbf{e}_{1j_1},\ldots,\textbf{e}_{nj_n}) = A_{j_1\cdots j_n}^1\,\textbf{b}_1 + \cdots +  A_{j_1\cdots j_n}^d\,\textbf{b}_d.

Then the scalars \{A_{j_1\cdots j_n}^k \mid 1\leq j_i\leq d_i, 1 \leq k \leq d\} completely determine the multilinear function f\!. In particular, if

\textbf{v}_i = \sum_{j=1}^{d_i} v_{ij} \textbf{e}_{ij}\!

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

f(\textbf{v}_1,\ldots,\textbf{v}_n) = \sum_{j_1=1}^{d_1} \cdots \sum_{j_n=1}^{d_n} \sum_{k=1}^{d} A_{j_1\cdots j_n}^k v_{1j_1}\cdots v_{nj_n} \textbf{b}_k.

Relation to tensor products[edit]

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

f\colon V_1 \times \cdots \times V_n \to W\text{,}

and linear maps

F\colon V_1 \otimes \cdots \otimes V_n \to W\text{,}

where V_1 \otimes \cdots \otimes V_n\! denotes the tensor product of V_1,\ldots,V_n. The relation between the functions f\! and F\! is given by the formula

F(v_1\otimes \cdots \otimes v_n) = f(v_1,\ldots,v_n).

Multilinear functions on n×n matrices[edit]

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 ≤ 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 \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\left(\sum_{j=1}^n A(1,j)\hat{e}_{j}, a_2, \ldots, a_n\right)
       = \sum_{j=1}^n A(1,j) D(\hat{e}_{j},a_2,\ldots,a_n)

Continuing this substitution for each a_i 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(\hat{e}_{k_{1}},\dots,\hat{e}_{k_{n}})
where, since in our case  1 \le i \le n

 \sum_{1\le k_i \le n} = \sum_{1\le k_1 \le n} \ldots \sum_{1\le k_i \le n} \ldots \sum_{1\le k_n \le n} \,
as a series of nested summations.

Therefore, D(A) is uniquely determined by how D operates on \hat{e}_{k_{1}},\dots,\hat{e}_{k_{n}}.

Example[edit]

In the case of 2×2 matrices we get


D(A) = A_{1,1}A_{2,1}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[edit]

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[edit]

References[edit]

  1. ^ Lang. Algebra. Springer; 3rd edition (January 8, 2002)