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 : V 1 × ⋯ × V n → W , where V 1 , … , V n and W are vector spaces (or modules over a commutative ring), with the following property: for each i , if all of the variables but v i are held constant, then f ( v 1 , … , v n ) is a linear function of v i .
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.
Any bilinear map is a multilinear map. For example, any inner product on a vector space is a multilinear map, as is the cross product of vectors in R 3 .The determinant of a matrix is an antisymmetric multilinear function of the columns (or rows) of a square matrix.If F : R m → R n is a Ck function, then the k th derivative of F at each point p in its domain can be viewed as a symmetric k -linear function D k f ( p ) : R m × ⋯ × R m → R n .The tensor-to-vector projection in multilinear subspace learning is a multilinear map as well.Let
f : V 1 × ⋯ × V n → W , 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 { e i 1 , … , e i d i } for each V i and a basis { b 1 , … , b d } for W (using bold for vectors), then we can define a collection of scalars A j 1 ⋯ j n k by
f ( e 1 j 1 , … , e n j n ) = A j 1 ⋯ j n 1 b 1 + ⋯ + A j 1 ⋯ j n d b d . Then the scalars { A j 1 ⋯ j n k ∣ 1 ≤ j i ≤ d i , 1 ≤ k ≤ d } completely determine the multilinear function f . In particular, if
v i = ∑ j = 1 d i v i j e i j for 1 ≤ i ≤ n , then
f ( v 1 , … , v n ) = ∑ j 1 = 1 d 1 ⋯ ∑ j n = 1 d n ∑ k = 1 d A j 1 ⋯ j n k v 1 j 1 ⋯ v n j n b k . Let's take a trilinear function
f : R 2 × R 2 × R 2 → R , where Vi = R2, di = 2, i = 1,2,3, and W = R, d = 1.
A basis for each Vi is { e i 1 , … , e i d i } = { e 1 , e 2 } = { ( 1 , 0 ) , ( 0 , 1 ) } . Let
f ( e 1 i , e 2 j , e 3 k ) = f ( e i , e j , e k ) = A i j k , where i , j , k ∈ { 1 , 2 } . In other words, the constant A i j k is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three V i ), namely:
{ e 1 , e 1 , e 1 } , { e 1 , e 1 , e 2 } , { e 1 , e 2 , e 1 } , { e 1 , e 2 , e 2 } , { e 2 , e 1 , e 1 } , { e 2 , e 1 , e 2 } , { e 2 , e 2 , e 1 } , { e 2 , e 2 , e 2 } . Each vector v i ∈ V i = R 2 can be expressed as a linear combination of the basis vectors
v i = ∑ j = 1 2 v i j e i j = v i 1 × e 1 + v i 2 × e 2 = v i 1 × ( 1 , 0 ) + v i 2 × ( 0 , 1 ) . The function value at an arbitrary collection of three vectors v i ∈ R 2 can be expressed as
f ( v 1 , v 2 , v 3 ) = ∑ i = 1 2 ∑ j = 1 2 ∑ k = 1 2 A i j k v 1 i v 2 j v 3 k . Or, in expanded form as
f ( ( a , b ) , ( c , d ) , ( e , f ) ) = a c e × f ( e 1 , e 1 , e 1 ) + a c f × f ( e 1 , e 1 , e 2 ) + a d e × f ( e 1 , e 2 , e 1 ) + a d f × f ( e 1 , e 2 , e 2 ) + b c e × f ( e 2 , e 1 , e 1 ) + b c f × f ( e 2 , e 1 , e 2 ) + b d e × f ( e 2 , e 2 , e 1 ) + b d f × f ( e 2 , e 2 , e 2 ) . There is a natural one-to-one correspondence between multilinear maps
f : V 1 × ⋯ × V n → W , and linear maps
F : V 1 ⊗ ⋯ ⊗ V n → W , where V 1 ⊗ ⋯ ⊗ V n denotes the tensor product of V 1 , … , V n . The relation between the functions f and F is given by the formula
F ( v 1 ⊗ ⋯ ⊗ v n ) = f ( v 1 , … , v n ) . 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 ai, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as
D ( A ) = D ( a 1 , … , a n ) , satisfying
D ( a 1 , … , c a i + a i ′ , … , a n ) = c D ( a 1 , … , a i , … , a n ) + D ( a 1 , … , a i ′ , … , a n ) . If we let e ^ j represent the jth row of the identity matrix, we can express each row ai as the sum
a i = ∑ j = 1 n A ( i , j ) e ^ j . Using the multilinearity of D we rewrite D(A) as
D ( A ) = D ( ∑ j = 1 n A ( 1 , j ) e ^ j , a 2 , … , a n ) = ∑ j = 1 n A ( 1 , j ) D ( e ^ j , a 2 , … , a n ) . Continuing this substitution for each ai we get, for 1 ≤ i ≤ n,
D ( A ) = ∑ 1 ≤ k i ≤ n A ( 1 , k 1 ) A ( 2 , k 2 ) … A ( n , k n ) D ( e ^ k 1 , … , e ^ k n ) , where, since in our case 1 ≤ i ≤ n,
is a series of nested summations.
Therefore, D(A) is uniquely determined by how D operates on e ^ k 1 , … , e ^ k n .
In the case of 2×2 matrices we get
D ( A ) = A 1 , 1 A 2 , 1 D ( e ^ 1 , e ^ 1 ) + A 1 , 1 A 2 , 2 D ( e ^ 1 , e ^ 2 ) + A 1 , 2 A 2 , 1 D ( e ^ 2 , e ^ 1 ) + A 1 , 2 A 2 , 2 D ( e ^ 2 , e ^ 2 ) Where e ^ 1 = [ 1 , 0 ] and e ^ 2 = [ 0 , 1 ] . If we restrict D to be an alternating function then D ( e ^ 1 , e ^ 1 ) = D ( e ^ 2 , e ^ 2 ) = 0 and D ( e ^ 2 , e ^ 1 ) = − D ( e ^ 1 , 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 . A multilinear map has a value of zero whenever one of its arguments is zero. 