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.
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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.
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 <math>a_i</math>, 1 ≤ i ≤ n be the rows of A. Then the multilinear function D can be written as
- <math>D(A) = D(a_{1},\ldots,a_{n}) \,</math>
satisfying
- <math>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}) \,</math>
If we let <math>\varepsilon_j</math> represent the jth row of the identity matrix we can express each row <math>a_{i}</math> as the sum
- <math>a_{i} = \sum_{j=1}^n A(i,j)\varepsilon_{j}</math>
Using the multilinearity of D we rewrite D(A) as
- <math>
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)
</math>
Continuing this substitution for each <math>a_i</math> we get, for 1 ≤ i ≤ n
- <math>
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}}) </math>
So D(A) is uniquely determined by how it operates on <math>D(\varepsilon_{k_{1}},\dots,\varepsilon_{k_{n}})</math>.
In the case of 2×2 matrices we get
- <math>
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) \, </math>
Where <math>\varepsilon_1 = [1,0]</math> and <math>\varepsilon_2 = [0,1]</math>. If we restrict D to be an alternating function then <math>D(\varepsilon_1,\varepsilon_1) = D(\varepsilon_2,\varepsilon_2) = 0</math> and <math>D(\varepsilon_2,\varepsilon_1) = -D(\varepsilon_1,\varepsilon_2) = -D(I)</math>. Letting <math>D(I) = 1</math> we get the determinant function on 2×2 matrices:
- <math>
D(A) = A_{1,1}A_{2,2} - A_{1,2}A_{2,1} \, </math>
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.