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

<math>f\colon V_1 \times \cdots \times V_n \to W\text{,}</math>

where <math>V_1,\ldots,V_n</math> (<math>n\in\mathbb Z_{\ge0}</math>) and <math>W</math> are vector spaces (or modules over a commutative ring), with the following property: for each <math>i</math>, if all of the variables but <math>v_i</math> are held constant, then <math>f(v_1, \ldots,

v_i, \ldots, v_n)</math> is a linear function of <math>v_i</math>.<ref>Template:Cite book</ref> 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 <math>2^2</math>.

A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer <math>k</math>, 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 <math>\mathbb R</math>-vector space is a multilinear map, as is the cross product of vectors in <math>\mathbb{R}^3</math>.
• 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 <math>F\colon \mathbb{R}^m \to \mathbb{R}^n</math> is a C^k function, then the <math>k</math>th derivative of <math>F</math> at each point <math>p</math> in its domain can be viewed as a symmetric <math>k</math>-linear function <math>D^k\!F\colon \mathbb{R}^m\times\cdots\times\mathbb{R}^m \to \mathbb{R}^n</math>.Template:Citation needed

Coordinate representation

Let

<math>f\colon V_1 \times \cdots \times V_n \to W\text{,}</math>

be a multilinear map between finite-dimensional vector spaces, where <math>V_i\!</math> has dimension <math>d_i\!</math>, and <math>W\!</math> has dimension <math>d\!</math>. If we choose a basis <math>\{\textbf{e}_{i1},\ldots,\textbf{e}_{id_i}\}</math> for each <math>V_i\!</math> and a basis <math>\{\textbf{b}_1,\ldots,\textbf{b}_d\}</math> for <math>W\!</math> (using bold for vectors), then we can define a collection of scalars <math>A_{j_1\cdots j_n}^k</math> by

<math>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.</math>

Then the scalars <math>\{A_{j_1\cdots j_n}^k \mid 1\leq j_i\leq d_i, 1 \leq k \leq d\}</math> completely determine the multilinear function <math>f\!</math>. In particular, if

<math>\textbf{v}_i = \sum_{j=1}^{d_i} v_{ij} \textbf{e}_{ij}\!</math>

for <math>1 \leq i \leq n\!</math>, then

<math>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.</math>

Example

Let's take a trilinear function

<math>g\colon R^2 \times R^2 \times R^2 \to R, </math>

where Template:Math, and Template:Math.

A basis for each Template:Mvar is <math>\{\textbf{e}_{i1},\ldots,\textbf{e}_{id_i}\} = \{\textbf{e}_{1}, \textbf{e}_{2}\} = \{(1,0), (0,1)\}.</math> Let

<math>g(\textbf{e}_{1i},\textbf{e}_{2j},\textbf{e}_{3k}) = f(\textbf{e}_{i},\textbf{e}_{j},\textbf{e}_{k}) = A_{ijk},</math>

where <math>i,j,k \in \{1,2\}</math>. In other words, the constant <math>A_{i j k}</math> is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three <math>V_i</math>), namely:

<math>

\{\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\}. </math>

Each vector <math>\textbf{v}_i \in V_i = R^2</math> can be expressed as a linear combination of the basis vectors

<math>\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).</math>

The function value at an arbitrary collection of three vectors <math>\textbf{v}_i \in R^2</math> can be expressed as

<math>g(\textbf{v}_1,\textbf{v}_2, \textbf{v}_3) = \sum_{i=1}^{2} \sum_{j=1}^{2} \sum_{k=1}^{2} A_{i j k} v_{1i} v_{2j} v_{3k},</math>

or in expanded form as

<math> \begin{align}

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{align} </math>

Relation to tensor products

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

<math>f\colon V_1 \times \cdots \times V_n \to W\text{,}</math>

and linear maps

<math>F\colon V_1 \otimes \cdots \otimes V_n \to W\text{,}</math>

where <math>V_1 \otimes \cdots \otimes V_n\!</math> denotes the tensor product of <math>V_1,\ldots,V_n</math>. The relation between the functions <math>f</math> and <math>F</math> is given by the formula

<math>f(v_1,\ldots,v_n)=F(v_1\otimes \cdots \otimes v_n).</math>

Multilinear functions on n×n matrices

One can consider multilinear functions, on an Template:Math matrix over a commutative ring Template:Mvar with identity, as a function of the rows (or equivalently the columns) of the matrix. Let Template:Math be such a matrix and Template:Math, be the rows of Template:Math. Then the multilinear function Template:Math can be written as

<math>D(A) = D(a_{1},\ldots,a_{n}),</math>

satisfying

<math>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}).</math>

If we let <math>\hat{e}_j</math> represent the Template:Mvarth row of the identity matrix, we can express each row Template:Math as the sum

<math>a_{i} = \sum_{j=1}^n A(i,j)\hat{e}_{j}.</math>

Using the multilinearity of Template:Math we rewrite Template:Math as

<math>

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).

</math>

Continuing this substitution for each Template:Math we get, for Template:Math,

<math>

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})\dots A(n,k_{n}) D(\hat{e}_{k_{1}},\dots,\hat{e}_{k_{n}}). </math>

Therefore, Template:Math is uniquely determined by how Template:Mvar operates on <math>\hat{e}_{k_{1}},\dots,\hat{e}_{k_{n}}</math>.

Example

In the case of 2×2 matrices, we get

<math>

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), \, </math>

where <math>\hat{e}_1 = [1,0]</math> and <math>\hat{e}_2 = [0,1]</math>. If we restrict <math>D</math> to be an alternating function, then <math>D(\hat{e}_1,\hat{e}_1) = D(\hat{e}_2,\hat{e}_2) = 0</math> and <math>D(\hat{e}_2,\hat{e}_1) = -D(\hat{e}_1,\hat{e}_2) = -D(I)</math>. Letting <math>D(I) = 1</math>, we get the determinant function on 2×2 matrices:

<math> D(A) = A_{1,1}A_{2,2} - A_{1,2}A_{2,1} .</math>

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

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