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
where formula_2 and formula_3 are vector spaces (or modules over a commutative ring), with the following property: for each formula_4, if all of the variables but formula_5 are held constant, then formula_6 is a linear function of formula_5.
A multilinear map of one variable is a linear map, and 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.
Coordinate representation.
Let
be a multilinear map between finite-dimensional vector spaces, where formula_16 has dimension formula_17, and formula_18 has dimension formula_19. If we choose a basis formula_20 for each formula_16 and a basis formula_22 for formula_18 (using bold for vectors), then we can define a collection of scalars formula_24 by
Then the scalars formula_26 completely determine the multilinear function formula_27. In particular, if
for formula_29, then
Example.
Let's take a trilinear function
where , and .
A basis for each is formula_32 Let
where formula_34. In other words, the constant formula_35 is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three formula_36), namely: 
Each vector formula_38 can be expressed as a linear combination of the basis vectors
The function value at an arbitrary collection of three vectors formula_40 can be expressed as
Or, in expanded form as
Relation to tensor products.
There is a natural one-to-one correspondence between multilinear maps
and linear maps
where formula_45 denotes the tensor product of formula_2. The relation between the functions formula_27 and formula_11 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
If we let formula_52 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 ,
where, since in our case ,
is a series of nested summations.
Therefore, is uniquely determined by how operates on formula_57.
Example.
In the case of 2×2 matrices we get
Where formula_59 and formula_60. If we restrict formula_61 to be an alternating function then formula_62 and formula_63. Letting formula_64 we get the determinant function on 2×2 matrices: