Chapter 101. Multilinear Maps

Multilinear maps generalize linear maps to functions with several vector inputs. A linear map has one vector argument. A bilinear map has two vector arguments. A multilinear map has any finite number of vector arguments and is linear in each argument separately.

This chapter gives the language needed for tensor products, exterior algebra, symmetric algebra, determinants, differential forms, and higher-order tensors. The main principle is simple: multilinear maps are nonlinear as functions of all variables together, but linear in each variable when the others are fixed. Standard references formulate tensor products through this same universal correspondence between multilinear maps and linear maps from tensor products.

101.1 Linear Maps Recalled

Let $V$ and $W$ be vector spaces over a field $F$ .

A function

T : V \to W

is linear if

T(u+v)=T(u)+T(v)

and

T(cv)=cT(v)

for all $u,v\in V$ and all $c\in F$ .

Equivalently,

T(au+bv)=aT(u)+bT(v).

Linearity says that $T$ preserves linear combinations.

Multilinearity applies this same condition to several inputs, one input at a time.

101.2 Definition of a Multilinear Map

Let

V_1,V_2,\ldots,V_k,W

be vector spaces over $F$ .

A map

A : V_1\times V_2\times \cdots \times V_k \to W

is multilinear if it is linear in each argument separately.

That means: for each index $i$ , if all variables except the $i$ -th one are held fixed, then the function

v_i \mapsto A(v_1,\ldots,v_i,\ldots,v_k)

is linear.

Explicitly,

A(v_1,\ldots,au_i+bw_i,\ldots,v_k) = aA(v_1,\ldots,u_i,\ldots,v_k) + bA(v_1,\ldots,w_i,\ldots,v_k).

This must hold for every argument position $i$ , every pair $u_i,w_i\in V_i$ , and every pair of scalars $a,b\in F$ .

A multilinear map with $k$ inputs is also called a $k$ -linear map.

101.3 Special Cases

The cases $k=1$ , $k=2$ , and $k=3$ have special names.

Number of inputs	Name	Form
1	Linear map	$V\to W$
2	Bilinear map	$V_1\times V_2\to W$
3	Trilinear map	$V_1\times V_2\times V_3\to W$
$k$	Multilinear map	$V_1\times\cdots\times V_k\to W$

Thus multilinear maps extend the familiar notion of a linear map.

101.4 Bilinear Maps

A bilinear map is a map

B : V\times W \to U

such that

B(av_1+bv_2,w) = aB(v_1,w)+bB(v_2,w)

and

B(v,aw_1+bw_2) = aB(v,w_1)+bB(v,w_2).

Examples include:

Map	Formula	Type
Dot product	$(u,v)\mapsto u\cdot v$	$\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$
Matrix multiplication	$(A,B)\mapsto AB$	$M_{m,n}\times M_{n,p}\to M_{m,p}$
Polynomial multiplication	$(p,q)\mapsto pq$	$P_m\times P_n\to P_{m+n}$
Outer product	$(u,v)\mapsto uv^T$	$\mathbb{R}^m\times\mathbb{R}^n\to M_{m,n}$

The dot product is bilinear over real vector spaces. Matrix multiplication is bilinear because it distributes over addition and respects scalar multiplication in each factor.

101.5 Multilinearity Versus Linearity

A multilinear map is generally not linear as a map from the product space.

For example, the ordinary product

B(x,y)=xy

is bilinear as a map

\mathbb{R}\times\mathbb{R}\to\mathbb{R}.

It is linear in $x$ when $y$ is fixed, and linear in $y$ when $x$ is fixed.

But it is not linear in the pair $(x,y)$ . Indeed,

B((x_1,y_1)+(x_2,y_2)) = (x_1+x_2)(y_1+y_2).

Expanding gives

x_1y_1+x_1y_2+x_2y_1+x_2y_2.

This differs from

B(x_1,y_1)+B(x_2,y_2) = x_1y_1+x_2y_2.

The extra cross terms show why multilinearity must be understood one variable at a time.

101.6 Coordinate Formula

Let

A : V_1\times\cdots\times V_k\to W

be multilinear.

Suppose each $V_i$ has a basis

e^{(i)}_1,\ldots,e^{(i)}_{n_i}.

v_i=\sum_{j_i=1}^{n_i} a^{(i)}_{j_i}e^{(i)}_{j_i},

then multilinearity gives

A(v_1,\ldots,v_k) = \sum_{j_1,\ldots,j_k} a^{(1)}_{j_1}\cdots a^{(k)}_{j_k} A(e^{(1)}_{j_1},\ldots,e^{(k)}_{j_k}).

Thus a multilinear map is determined completely by its values on basis tuples.

This is the higher-order version of the fact that a linear map is determined by its values on a basis.

101.7 Components

When $W=F$ , a multilinear map

A : V_1\times\cdots\times V_k\to F

is called a multilinear form.

Its values on basis tuples are scalars:

A(e^{(1)}_{j_1},\ldots,e^{(k)}_{j_k}).

These scalars form a multidimensional array of components.

For $k=2$ , the component array is a matrix.

For $k=3$ , the component array has three indices.

For general $k$ , the component array has $k$ indices.

This is one reason tensors are often represented computationally as multidimensional arrays.

101.8 Multilinear Forms

A multilinear form is a scalar-valued multilinear map.

Examples include:

Form	Number of inputs	Meaning
Linear functional	1	Measures one vector
Bilinear form	2	Pairs two vectors
Inner product	2	Measures angle and length
Determinant	$n$	Measures signed volume
Alternating form	$k$	Measures oriented $k$ -volume

The determinant is the most important classical example.

For vectors $v_1,\ldots,v_n\in F^n$ , the function

(v_1,\ldots,v_n) \mapsto \det[v_1\ \cdots\ v_n]

is multilinear in the column vectors.

It is also alternating, since swapping two columns changes the sign.

101.9 Multilinear Maps and Tensor Products

Tensor products are designed to represent multilinear maps as linear maps.

Let

A : V_1\times\cdots\times V_k\to W

be multilinear.

Then there is a unique linear map

\widetilde{A} : V_1\otimes\cdots\otimes V_k \to W

such that

\widetilde{A}(v_1\otimes\cdots\otimes v_k) = A(v_1,\ldots,v_k).

This is the universal property of the tensor product.

It says that studying multilinear maps out of a product is equivalent to studying linear maps out of a tensor product. This correspondence is a standard formulation of the tensor product.

101.10 Space of Multilinear Maps

The set of all multilinear maps

V_1\times\cdots\times V_k\to W

is itself a vector space.

If $A$ and $B$ are multilinear, then

(A+B)(v_1,\ldots,v_k) = A(v_1,\ldots,v_k)+B(v_1,\ldots,v_k).

If $c\in F$ , then

(cA)(v_1,\ldots,v_k) = cA(v_1,\ldots,v_k).

Both operations preserve multilinearity.

This vector space is often denoted

\operatorname{Mult}(V_1,\ldots,V_k;W).

When $W=F$ , the space of multilinear forms is naturally related to

V_1^*\otimes\cdots\otimes V_k^*.

101.11 Dimension Count

Assume all vector spaces are finite-dimensional.

Let

\dim V_i=n_i

and

\dim W=m.

A multilinear map is determined by its values on all basis tuples.

There are

n_1n_2\cdots n_k

basis tuples.

For each tuple, the value lies in $W$ , an $m$ -dimensional space.

Therefore,

\dim \operatorname{Mult}(V_1,\ldots,V_k;W) = m n_1n_2\cdots n_k.

For scalar-valued multilinear forms,

\dim \operatorname{Mult}(V_1,\ldots,V_k;F) = n_1n_2\cdots n_k.

101.12 Alternating Multilinear Maps

A multilinear map

A : V^k\to W

is alternating if

A(v_1,\ldots,v_k)=0

whenever two arguments are equal.

Equivalently, over fields of characteristic not equal to $2$ , swapping two arguments changes the sign:

A(\ldots,v_i,\ldots,v_j,\ldots) = - A(\ldots,v_j,\ldots,v_i,\ldots).

The determinant is alternating.

Exterior algebra is built to represent alternating multilinear maps, just as tensor algebra represents general multilinear maps.

101.13 Symmetric Multilinear Maps

A multilinear map

A : V^k\to W

is symmetric if its value remains unchanged under every permutation of the arguments:

A(v_{\sigma(1)},\ldots,v_{\sigma(k)}) = A(v_1,\ldots,v_k)

for every permutation $\sigma\in S_k$ .

Examples include symmetric bilinear forms:

B(u,v)=B(v,u).

The Hessian of a sufficiently smooth scalar function gives a symmetric bilinear form at each point.

Symmetric algebra represents symmetric multilinear structure.

101.14 Tensor Fields and Multilinear Maps

In geometry, tensors are often defined as multilinear maps on vectors and covectors.

For example, a tensor of type $(r,s)$ on a vector space $V$ may be viewed as a multilinear map taking $r$ covector inputs and $s$ vector inputs:

T : (V^*)^r\times V^s \to F.

This coordinate-free definition is important because it describes the tensor independently of any chosen basis.

Coordinate arrays are representations of tensors, not the tensors themselves.

101.15 Currying

A multilinear map can be partially evaluated.

A : V_1\times V_2\times V_3\to W

is trilinear and $v_1\in V_1$ is fixed, then

(v_2,v_3) \mapsto A(v_1,v_2,v_3)

is bilinear.

This process is called partial evaluation.

It allows multilinear maps to be studied recursively.

A bilinear map

B : V\times W\to U

may also be viewed as a linear map

V\to \operatorname{Hom}(W,U)

by sending $v$ to the linear map

w\mapsto B(v,w).

This transformation of arguments is called currying.

101.16 Pullback of Multilinear Forms

Let

T_i : U_i\to V_i

be linear maps, and let

A : V_1\times\cdots\times V_k\to F

be a multilinear form.

The pullback of $A$ along the maps $T_i$ is the multilinear form

(T_1,\ldots,T_k)^*A : U_1\times\cdots\times U_k\to F

defined by

((T_1,\ldots,T_k)^*A)(u_1,\ldots,u_k) = A(T_1u_1,\ldots,T_ku_k).

Pullback is the natural way multilinear forms change under linear maps.

In geometry, this idea extends to differential forms and smooth maps between manifolds.

101.17 Example: Matrix Multiplication

Matrix multiplication is bilinear.

Let

A_1,A_2\in M_{m,n}, \qquad B\in M_{n,p}, \qquad c\in F.

Then

(A_1+A_2)B = A_1B+A_2B

and

(cA_1)B = c(A_1B).

Similarly,

A(B_1+B_2) = AB_1+AB_2

and

A(cB_1) = c(AB_1).

Thus

(A,B)\mapsto AB

is bilinear.

This example is important because it shows that multilinearity appears in ordinary matrix algebra, not only in abstract tensor theory.

101.18 Example: Determinant

The determinant

\det : (F^n)^n\to F

takes $n$ column vectors and returns a scalar.

It is multilinear in its columns.

For $n=2$ , if

u= \begin{bmatrix} a \\ b \end{bmatrix}, \qquad v= \begin{bmatrix} c \\ d \end{bmatrix},

then

\det[u\ v] = ad-bc.

If $u$ is replaced by $u_1+u_2$ , the determinant splits into the sum of two determinants. If $u$ is multiplied by $c$ , the determinant is multiplied by $c$ .

The same holds in every column.

This multilinearity is one of the structural reasons determinants are useful.

101.19 Example: A Trilinear Map

Let

A : \mathbb{R}^3\times\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R}

be defined by

A(u,v,w)=u\cdot(v\times w).

This is the scalar triple product.

It measures signed volume.

The map is trilinear. It is linear in $u$ because the dot product is linear in $u$ . It is linear in $v$ and $w$ because the cross product is bilinear.

It is also alternating. If two vectors are equal, the volume collapses to zero.

101.20 Computational Representation

In computation, a scalar-valued $k$ -linear map on

F^{n_1}\times\cdots\times F^{n_k}

can be stored as an array

A_{i_1,\ldots,i_k}.

Then

A(v_1,\ldots,v_k) = \sum_{i_1,\ldots,i_k} A_{i_1,\ldots,i_k} (v_1)_{i_1}\cdots(v_k)_{i_k}.

For vector-valued maps into $F^m$ , one additional output index is used:

A^\alpha_{i_1,\ldots,i_k}.

The output components are

A(v_1,\ldots,v_k)^\alpha = \sum_{i_1,\ldots,i_k} A^\alpha_{i_1,\ldots,i_k} (v_1)_{i_1}\cdots(v_k)_{i_k}.

This notation is the bridge between abstract multilinear algebra and tensor computation.

101.21 Summary

Multilinear maps are functions with several vector inputs that are linear in each input separately.

Concept	Meaning
Linear map	One vector input
Bilinear map	Two vector inputs
Multilinear map	Several vector inputs
Multilinear form	Scalar-valued multilinear map
Alternating map	Changes sign under swaps
Symmetric map	Invariant under permutations

The central structural fact is that multilinear maps from

V_1\times\cdots\times V_k

correspond to linear maps from

V_1\otimes\cdots\otimes V_k.

This correspondence explains why tensor products are the natural language for multilinear algebra. Multilinear maps give the operational viewpoint; tensor products give the linear representation.