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 a 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.
Examples
- An inner product (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.
- Bilinear maps are multilinear maps.
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
General discussion of where this leads is at multilinear algebra.