Multilinear map explained

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\colon V _{1} x … x V _{n} \to W ,

where

V _{1,\ldots,V} _{n} (n\in Z _{\ge0} ) 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,} \ldots, v _{i,} \ldots, v _{n)} is a linear function of 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} .

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 , 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 on a

R -vector space is a multilinear map, as is the cross product of vectors in R ^{3} .

• The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
• If

F\colon R ^{m} \to R ^{n} is a C^k 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 ^{kF\colon} R ^{m x … x R} ^{m} \to R ^{n} .

Coordinate representation

Let

f\colon V _{1} x … x V _{n} \to 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 \{ bf{e} _{i1} ,\ldots, bf{e} _{id _{i}} \} for each V _{i} and a basis \{ bf{b} _{1,\ldots,bf{b}} _{d\}} for W (using bold for vectors), then we can define a collection of scalars A ^{k} _{j _{1 …} j _{n}} byf( bf{e} _{1 j _{1}} ,\ldots, bf{e} _{nj _{n}} ) = A ^{1 bf{b}} _{1} + … + A ^{d bf{b}} _{d.}

Then the scalars

\{A ^{k} _{j _{1 …} j _{n}} \mid 1 \leq j _{i\leq} d _{i,} 1 \leq k \leq d\} completely determine the multilinear function f . In particular, ifbf{v} _{i} = \sum ^{d _{i}} _{j =1} v _{ij} bf{e} _{ij}

for

1 \leq i \leq n , thenf( bf{v} _{1,\ldots,bf{v}} _{n)} = \sum ^{d _{1}} _{j _{1=1}} … \sum ^{d _{n}} _{j _{n=1}} \sum ^{d} _{k =1} A ^{k} _{j _{1 …} j _{n}} v _{1 j _{1}} … v _{nj _{n}} bf{b} _{k.}

Example

Let's take a trilinear function

g\colon R ^{2} x R ^{2} x R ^{2} \to R, where, and .

A basis for each is

\{ bf{e} _{i1} ,\ldots, bf{e} _{id _{i}} \} = \{ bf{e} _{1} , bf{e} _{2} \} = \{( 1 , 0 ), ( 0 , 1 )\}. Letg( bf{e} _{1i} , bf{e} _{2j} , bf{e} _{3k} ) = f( bf{e} _{i} , bf{e} _{j} , bf{e} _{k} ) = A _{ijk} , where i,j,k \in \{ 1 , 2 \} . In other words, the constant A _{i} 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: \{ bf{e} _{1,} bf{e} _{1,} bf{e} _{1\},} \{ bf{e} _{1,} bf{e} _{1,} bf{e} _{2\},} \{ bf{e} _{1,} bf{e} _{2,} bf{e} _{1\}, \{bf{e}} _{1,} bf{e} _{2,} bf{e} _{2\}, \{bf{e}} _{2,} bf{e} _{1,} bf{e} _{1\},} \{ bf{e} _{2,} bf{e} _{1,} bf{e} _{2\},} \{ bf{e} _{2,} bf{e} _{2,} bf{e} _{1\}, \{bf{e}} _{2,} bf{e} _{2,} bf{e} _{2\}.}

Each vector

bf{v} _{i} \in V _{i} = R ^{2} can be expressed as a linear combination of the basis vectorsbf{v} _{i} = \sum ^{2} _{j =1} v _{ij} bf{e} _{ij} = v _{i1} x bf{e} _{1} + v _{i2} x bf{e} _{2} = v _{i1} x ( 1 , 0 ) + v _{i2} x ( 0 , 1 ).

The function value at an arbitrary collection of three vectors

bf{v} _{i} \in R ^{2} can be expressed asg( bf{v} _{1,bf{v}} _{2,} bf{v} _{3)} = \sum ^{2} _{i =1} \sum ^{2} _{j =1} \sum ^{2} _{k =1} A _{i} v _{1i} v _{2j} v _{3k} , or in expanded form as\begin{align} g((a,b),(c,d)&, (e,f)) = ace x g( bf{e} _{1,} bf{e} _{1,} bf{e} _{1)} + acf x g( bf{e} _{1,} bf{e} _{1,} bf{e} _{2)} \\ & + ade x g( bf{e} _{1,} bf{e} _{2,} bf{e} _{1)} + adf x g( bf{e} _{1,} bf{e} _{2,} bf{e} _{2)} + bce x g( bf{e} _{2,} bf{e} _{1,} bf{e} _{1)} + bcf x g( bf{e} _{2,} bf{e} _{1,} bf{e} _{2)} \ & + bde x g( bf{e} _{2,} bf{e} _{2,} bf{e} _{1)} + bdf x g( bf{e} _{2,} bf{e} _{2,} bf{e} _{2). \end{align}}

Relation to tensor products

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

f\colon V _{1} x … x V _{n} \to W ,

and linear maps

F\colon V _{1} ⊗ … ⊗ V _{n} \to W ,

where

V _{1} ⊗ … ⊗ V _{n} denotes the tensor product of V _{1,\ldots,V} _{n} . The relation between the functions f and F is given by the formulaf(v _{1,\ldots,v} _{n)=F(v} _{1 ⊗} … ⊗ v _{n).}

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

D(A) = D(a _{1} ,\ldots,a _{n} ),

satisfying

D(a _{1} ,\ldots,c a _{i} + a _{i} ',\ldots,a _{n} ) = c D(a _{1} ,\ldots,a _{i} ,\ldots,a _{n} ) + D(a _{1} ,\ldots,a _{i} ',\ldots,a _{n} ).

If we let

\hat{e} _{j} represent the th row of the identity matrix, we can express each row as the suma _{i} = \sum ^{n} _{j =1} A(i,j)\hat{e} _{j} .

Using the multilinearity of we rewrite as

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

Continuing this substitution for each we get, for,

D(A) = \sum _{1 \le k _{1} \le n} \ldots \sum _{1 \le k _{i} \le n} \ldots \sum _{1 \le k _{n} \le n} A( 1 ,k _{1} )A( 2 ,k _{2} ) ... A(n,k _{n} ) D(\hat{e} _{k _{1}} , ... ,\hat{e} _{k _{n}} ).

Therefore, is uniquely determined by how operates on

\hat{e} _{k _{1}} , ... ,\hat{e} _{k _{n}} .

Example

In the case of 2×2 matrices, we get

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

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

Properties

• A multilinear map has a value of zero whenever one of its arguments is zero.

See also

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

References

1. Book: Lang, Serge . Serge Lang . Algebra . XIII. Matrices and Linear Maps §S Determinants . https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA511 . 2005 . 2002 . Springer . 3rd . 978-0-387-95385-4 . 511– . 211 . Graduate Texts in Mathematics.