Wikipedia / Multilinear map
Wikipedia / Multilinear map
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
are vector spaces (or moduless over a commutative ring is a linear function. One way to visualize this is to imagine two orthogonal vectors; if one of these vectors is scaled by a factor of 2 while the other remains unchanged, the cross product.
A multilinear map of one variable is a linear map, and of two variables is a bilinear map, 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 two coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.
Examples
Any bilinear map is a multilinear map. For example, any inner product-vector space is a multilinear map, as is the cross product.
The determinant of a square matrix is a multilinear function of the columns (or rows); it is also an alternating function of the columns (or rows).
is a Ck function in its domain can be viewed as a symmetric.
Coordinate representation
Let
be a multilinear map between finite-dimensional. If we choose a basis by
. In particular, if
, then
Example
Let's take a trilinear function
where , and .
A basis for each Let
can be expressed as a linear combination of the basis vectors
Relation to tensor products
There is a natural one-to-one correspondence between multilinear maps
and linear maps
denotes the tensor product is given by the formula
Multilinear functions on n×n matrices
One can consider multilinear functions, on an matrix over a commutative ring with identity, as a function of the rows (or equivalently the columns) of the matrix. Let be such a matrix and , be the rows of . Then the multilinear function can be written as
satisfying
represent the th row of the identity matrix, we can express each row as the sum
Using the multilinearity of we rewrite as
Continuing this substitution for each we get, for ,
Therefore, is uniquely determined by how .
Example
In the case of 22 matrices, we get
, we get the determinant function on 22 matrices:
Properties
A multilinear map has a value of zero whenever one of its arguments is zero.