Multilinear map

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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 , {\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}

where V 1 , … , V n {\displaystyle V_{1},\ldots ,V_{n}} ( n ∈ Z ≥ 0 {\displaystyle n\in \mathbb {Z} _{\geq 0}} ) and W {\displaystyle W} are vector spaces (or modules over a commutative ring), with the following property: for each i {\displaystyle i} , if all of the variables but v i {\displaystyle v_{i}} are held constant, then f ( v 1 , … , v i , … , v n ) {\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n})} is a linear function of v i {\displaystyle v_{i}} .^[1] 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 likewise scales by a factor of two. If both are scaled by a factor of 2, the cross product scales by a factor of 2 2 {\displaystyle 2^{2}} .

A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer k {\displaystyle k} , 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.

• Any bilinear map is a multilinear map. For example, any inner product on a R {\displaystyle \mathbb {R} } -vector space is a multilinear map, as is the cross product of vectors in R 3 {\displaystyle \mathbb {R} ^{3}} .
• 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).
• If F : R m → R n {\displaystyle F\colon \mathbb {R} ^{m}\to \mathbb {R} ^{n}} is a C^k function, then the k {\displaystyle k} th derivative of F {\displaystyle F} at each point p {\displaystyle p} in its domain can be viewed as a symmetric k {\displaystyle k} -linear function D k F : R m × ⋯ × R m → R n {\displaystyle D^{k}\!F\colon \mathbb {R} ^{m}\times \cdots \times \mathbb {R} ^{m}\to \mathbb {R} ^{n}} .^{[citation needed]}

Let

f : V 1 × ⋯ × V n → W , {\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}

be a multilinear map between finite-dimensional vector spaces, where V i {\displaystyle V_{i}\!} has dimension d i {\displaystyle d_{i}\!} , and W {\displaystyle W\!} has dimension d {\displaystyle d\!} . If we choose a basis { e i 1 , … , e i d i } {\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}} for each V i {\displaystyle V_{i}\!} and a basis { b 1 , … , b d } {\displaystyle \{{\textbf {b}}_{1},\ldots ,{\textbf {b}}_{d}\}} for W {\displaystyle W\!} (using bold for vectors), then we can define a collection of scalars A j 1 ⋯ j n k {\displaystyle A_{j_{1}\cdots 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 . {\displaystyle f({\textbf {e}}_{1j_{1}},\ldots ,{\textbf {e}}_{nj_{n}})=A_{j_{1}\cdots j_{n}}^{1}\,{\textbf {b}}_{1}+\cdots +A_{j_{1}\cdots j_{n}}^{d}\,{\textbf {b}}_{d}.}

Then the scalars { A j 1 ⋯ j n k ∣ 1 ≤ j i ≤ d i , 1 ≤ k ≤ d } {\displaystyle \{A_{j_{1}\cdots j_{n}}^{k}\mid 1\leq j_{i}\leq d_{i},1\leq k\leq d\}} completely determine the multilinear function f {\displaystyle f\!} . In particular, if

v i = ∑ j = 1 d i v i j e i j {\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{d_{i}}v_{ij}{\textbf {e}}_{ij}\!}

for 1 ≤ i ≤ n {\displaystyle 1\leq i\leq 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 . {\displaystyle f({\textbf {v}}_{1},\ldots ,{\textbf {v}}_{n})=\sum _{j_{1}=1}^{d_{1}}\cdots \sum _{j_{n}=1}^{d_{n}}\sum _{k=1}^{d}A_{j_{1}\cdots j_{n}}^{k}v_{1j_{1}}\cdots v_{nj_{n}}{\textbf {b}}_{k}.}

Let's take a trilinear function

g : R 2 × R 2 × R 2 → R , {\displaystyle g\colon R^{2}\times R^{2}\times R^{2}\to R,}

where V_i = R², d_i = 2, i = 1,2,3, and W = R, d = 1.

A basis for each V_i is { e i 1 , … , e i d i } = { e 1 , e 2 } = { ( 1 , 0 ) , ( 0 , 1 ) } . {\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}=\{{\textbf {e}}_{1},{\textbf {e}}_{2}\}=\{(1,0),(0,1)\}.} Let

g ( e 1 i , e 2 j , e 3 k ) = f ( e i , e j , e k ) = A i j k , {\displaystyle g({\textbf {e}}_{1i},{\textbf {e}}_{2j},{\textbf {e}}_{3k})=f({\textbf {e}}_{i},{\textbf {e}}_{j},{\textbf {e}}_{k})=A_{ijk},}

where i , j , k ∈ { 1 , 2 } {\displaystyle i,j,k\in \{1,2\}} . In other words, the constant A i j k {\displaystyle A_{ijk}} 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 {\displaystyle 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 } . {\displaystyle \{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}\}.}

Each vector v i ∈ V i = R 2 {\displaystyle {\textbf {v}}_{i}\in 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 ) . {\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{2}v_{ij}{\textbf {e}}_{ij}=v_{i1}\times {\textbf {e}}_{1}+v_{i2}\times {\textbf {e}}_{2}=v_{i1}\times (1,0)+v_{i2}\times (0,1).}

The function value at an arbitrary collection of three vectors v i ∈ R 2 {\displaystyle {\textbf {v}}_{i}\in R^{2}} can be expressed as

g ( 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 , {\displaystyle g({\textbf {v}}_{1},{\textbf {v}}_{2},{\textbf {v}}_{3})=\sum _{i=1}^{2}\sum _{j=1}^{2}\sum _{k=1}^{2}A_{ijk}v_{1i}v_{2j}v_{3k},}

or in expanded form as

g ( ( a , b ) , ( c , d ) , ( e , f ) ) = a c e × g ( e 1 , e 1 , e 1 ) + a c f × g ( e 1 , e 1 , e 2 ) + a d e × g ( e 1 , e 2 , e 1 ) + a d f × g ( e 1 , e 2 , e 2 ) + b c e × g ( e 2 , e 1 , e 1 ) + b c f × g ( e 2 , e 1 , e 2 ) + b d e × g ( e 2 , e 2 , e 1 ) + b d f × g ( e 2 , e 2 , e 2 ) . {\displaystyle {\begin{aligned}g((a,b),(c,d)&,(e,f))=ace\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1})+acf\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+ade\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1})+adf\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2})+bce\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1})+bcf\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+bde\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1})+bdf\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}).\end{aligned}}}

There is a natural one-to-one correspondence between multilinear maps

f : V 1 × ⋯ × V n → W , {\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}

and linear maps

F : V 1 ⊗ ⋯ ⊗ V n → W , {\displaystyle F\colon V_{1}\otimes \cdots \otimes V_{n}\to W{\text{,}}}

where V 1 ⊗ ⋯ ⊗ V n {\displaystyle V_{1}\otimes \cdots \otimes V_{n}\!} denotes the tensor product of V 1 , … , V n {\displaystyle V_{1},\ldots ,V_{n}} . The relation between the functions f {\displaystyle f} and F {\displaystyle F} is given by the formula

f ( v 1 , … , v n ) = F ( v 1 ⊗ ⋯ ⊗ v n ) . {\displaystyle f(v_{1},\ldots ,v_{n})=F(v_{1}\otimes \cdots \otimes 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 a_i, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as

D ( A ) = D ( a 1 , … , a n ) , {\displaystyle D(A)=D(a_{1},\ldots ,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 ) . {\displaystyle D(a_{1},\ldots ,ca_{i}+a_{i}',\ldots ,a_{n})=cD(a_{1},\ldots ,a_{i},\ldots ,a_{n})+D(a_{1},\ldots ,a_{i}',\ldots ,a_{n}).}

If we let e ^ j {\displaystyle {\hat {e}}_{j}} represent the jth row of the identity matrix, we can express each row a_i as the sum

a i = ∑ j = 1 n A ( i , j ) e ^ j . {\displaystyle a_{i}=\sum _{j=1}^{n}A(i,j){\hat {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 ) . {\displaystyle D(A)=D\left(\sum _{j=1}^{n}A(1,j){\hat {e}}_{j},a_{2},\ldots ,a_{n}\right)=\sum _{j=1}^{n}A(1,j)D({\hat {e}}_{j},a_{2},\ldots ,a_{n}).}

Continuing this substitution for each a_i we get, for 1 ≤ i ≤ n,

D ( A ) = ∑ 1 ≤ k 1 ≤ n … ∑ 1 ≤ k i ≤ n … ∑ 1 ≤ k n ≤ n A ( 1 , k 1 ) A ( 2 , k 2 ) … A ( n , k n ) D ( e ^ k 1 , … , e ^ k n ) . {\displaystyle D(A)=\sum _{1\leq k_{1}\leq n}\ldots \sum _{1\leq k_{i}\leq n}\ldots \sum _{1\leq k_{n}\leq n}A(1,k_{1})A(2,k_{2})\dots A(n,k_{n})D({\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}).}

Therefore, D(A) is uniquely determined by how D operates on e ^ k 1 , … , e ^ k n {\displaystyle {\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}} .

In the case of 2×2 matrices, we get

D ( A ) = A 1 , 1 A 1 , 2 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 ) , {\displaystyle D(A)=A_{1,1}A_{1,2}D({\hat {e}}_{1},{\hat {e}}_{1})+A_{1,1}A_{2,2}D({\hat {e}}_{1},{\hat {e}}_{2})+A_{1,2}A_{2,1}D({\hat {e}}_{2},{\hat {e}}_{1})+A_{1,2}A_{2,2}D({\hat {e}}_{2},{\hat {e}}_{2}),\,}

where e ^ 1 = [ 1 , 0 ] {\displaystyle {\hat {e}}_{1}=[1,0]} and e ^ 2 = [ 0 , 1 ] {\displaystyle {\hat {e}}_{2}=[0,1]} . If we restrict D {\displaystyle D} to be an alternating function, then D ( e ^ 1 , e ^ 1 ) = D ( e ^ 2 , e ^ 2 ) = 0 {\displaystyle D({\hat {e}}_{1},{\hat {e}}_{1})=D({\hat {e}}_{2},{\hat {e}}_{2})=0} and D ( e ^ 2 , e ^ 1 ) = − D ( e ^ 1 , e ^ 2 ) = − D ( I ) {\displaystyle D({\hat {e}}_{2},{\hat {e}}_{1})=-D({\hat {e}}_{1},{\hat {e}}_{2})=-D(I)} . Letting D ( I ) = 1 {\displaystyle 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 . {\displaystyle 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.

• Algebraic form
• Multilinear form
• Homogeneous polynomial
• Homogeneous function
• Tensors