Chapter 102. Bilinear Forms

A bilinear form is a scalar-valued bilinear map. It takes two vector inputs and returns a scalar, while remaining linear in each input separately.

Bilinear forms generalize dot products, inner products, area forms, symplectic forms, and quadratic forms. They provide a bridge between linear algebra, geometry, and tensor theory. In finite dimensions, a bilinear form is represented by a matrix once a basis is chosen. The same form may then be studied through algebraic equations, geometric orthogonality, and matrix invariants. A bilinear form on a vector space $V$ over a field $F$ is commonly defined as a map $V \times V \to F$ that is linear in each coordinate.

102.1 Definition

Let $V$ be a vector space over a field $F$ .

A bilinear form on $V$ is a map

B : V \times V \to F

such that $B$ is linear in each argument separately.

Thus, for all $u,u_1,u_2,v,v_1,v_2\in V$ and all scalars $a,b\in F$ ,

B(au_1+bu_2,v) = aB(u_1,v)+bB(u_2,v),

and

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

The first identity says that $B$ is linear in the first argument. The second says that $B$ is linear in the second argument.

The value $B(u,v)$ is often called the pairing of $u$ and $v$ .

102.2 Examples

The standard dot product on $\mathbb{R}^n$ is a bilinear form:

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

In coordinates,

B(u,v) = u_1v_1+\cdots+u_nv_n.

Another example on $\mathbb{R}^2$ is

B(u,v)=u_1v_2-u_2v_1.

u= \begin{bmatrix} u_1\\ u_2 \end{bmatrix}, \qquad v= \begin{bmatrix} v_1\\ v_2 \end{bmatrix},

then

B(u,v) = u_1v_2-u_2v_1.

This is the determinant of the $2\times 2$ matrix with columns $u$ and $v$ . It measures signed area.

The two examples behave differently. The dot product is symmetric. The signed area form changes sign when the two inputs are exchanged.

102.3 Matrix Representation

Let $V$ have basis

e_1,\ldots,e_n.

A bilinear form $B$ is determined by the scalars

b_{ij}=B(e_i,e_j).

These scalars form the matrix

A= \begin{bmatrix} b_{11} & b_{12} & \cdots & b_{1n}\\ b_{21} & b_{22} & \cdots & b_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ b_{n1} & b_{n2} & \cdots & b_{nn} \end{bmatrix}.

u= \sum_i x_i e_i, \qquad v= \sum_j y_j e_j,

then bilinearity gives

B(u,v) = \sum_{i,j} x_i y_j B(e_i,e_j).

In matrix notation,

B(u,v)=x^T A y.

Here $x$ and $y$ are the coordinate column vectors of $u$ and $v$ .

Thus every bilinear form on a finite-dimensional vector space is represented by a matrix.

102.4 Dependence on Basis

The matrix of a bilinear form depends on the chosen basis.

Suppose $A$ is the matrix of $B$ in one basis, and suppose a change of basis is given by an invertible matrix $P$ . Then the matrix in the new basis is

A' = P^T A P.

This transformation differs from similarity transformation

A' = P^{-1}AP.

Similarity is used for linear maps. Congruence,

A' = P^TAP,

is used for bilinear forms.

This distinction is important. Bilinear forms are classified by congruence, not by similarity.

102.5 Symmetric Bilinear Forms

A bilinear form $B$ is symmetric if

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

for all $u,v\in V$ .

In matrix form, this means

A^T=A.

The dot product is symmetric:

u\cdot v=v\cdot u.

Symmetric bilinear forms are central in geometry because they define notions of length, angle, orthogonality, and curvature.

If a symmetric bilinear form is positive definite over $\mathbb{R}$ , then it is an inner product.

102.6 Alternating Bilinear Forms

A bilinear form $B$ is alternating if

B(v,v)=0

for every $v\in V$ .

Alternating forms encode oriented area rather than length.

For example,

B(u,v)=u_1v_2-u_2v_1

is alternating because

B(u,u)=u_1u_2-u_2u_1=0.

Over fields whose characteristic is not $2$ , alternating forms satisfy

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

Thus swapping the inputs reverses the sign.

In matrix form, an alternating bilinear form has a skew-symmetric matrix with zero diagonal entries.

102.7 Skew-Symmetric Forms

A bilinear form $B$ is skew-symmetric if

B(u,v)=-B(v,u)

for all $u,v\in V$ .

Every alternating form is skew-symmetric. If the field has characteristic not equal to $2$ , the converse also holds. This distinction matters over fields of characteristic $2$ , where signs behave differently.

The matrix of a skew-symmetric form satisfies

A^T=-A.

For real vector spaces, skew-symmetric forms appear in mechanics, symplectic geometry, and Hamiltonian systems.

102.8 Orthogonality

A bilinear form defines an orthogonality relation.

Vectors $u,v\in V$ are orthogonal with respect to $B$ if

B(u,v)=0.

This generalizes the usual perpendicularity defined by the dot product.

However, for general bilinear forms, orthogonality may fail to be symmetric. It may happen that

B(u,v)=0

but

B(v,u)\ne 0.

For symmetric and alternating forms, orthogonality behaves more regularly.

If $S\subseteq V$ , its orthogonal complement is

S^\perp = \{v\in V : B(s,v)=0\text{ for all }s\in S\}.

For non-symmetric forms, one must distinguish left and right orthogonal complements.

102.9 Radical

The right radical of $B$ is

\operatorname{rad}_R(B) = \{v\in V : B(u,v)=0\text{ for all }u\in V\}.

The left radical is

\operatorname{rad}_L(B) = \{u\in V : B(u,v)=0\text{ for all }v\in V\}.

If $B$ is symmetric or alternating, these two radicals coincide. The common radical is denoted

\operatorname{rad}(B).

A vector in the radical is invisible to the form. It pairs to zero with every vector.

102.10 Nondegenerate Forms

A bilinear form is nondegenerate if its radical is zero.

Equivalently, for every nonzero $u\in V$ , there exists $v\in V$ such that

B(u,v)\ne 0.

In finite dimensions, if $A$ is the matrix of $B$ , then $B$ is nondegenerate exactly when

\det A\ne 0.

Thus nondegeneracy of a bilinear form corresponds to invertibility of its matrix. This condition does not depend on the chosen basis.

The standard dot product on $\mathbb{R}^n$ is nondegenerate. The zero form is maximally degenerate.

102.11 Bilinear Forms and Dual Spaces

Every bilinear form defines a linear map

\Phi_B : V \to V^*

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

Thus a vector $u$ is sent to the linear functional

v\mapsto B(u,v).

The form $B$ is nondegenerate exactly when $\Phi_B$ is an isomorphism in finite dimensions.

This identifies $V$ with its dual space $V^*$ , but the identification depends on $B$ . Different bilinear forms produce different identifications.

102.12 Quadratic Forms

A quadratic form is a function

Q : V \to F

that can be written as

Q(v)=B(v,v)

for some bilinear form $B$ .

If $B$ is represented by a matrix $A$ , then

Q(x)=x^TAx.

Only the symmetric part of $A$ contributes to $Q$ when the field has characteristic not equal to $2$ . Indeed,

x^TAx = x^T\left(\frac{A+A^T}{2}\right)x.

Thus quadratic forms are naturally associated with symmetric bilinear forms in the usual real and complex settings.

Examples include:

Quadratic form	Matrix
$x^2+y^2$	$\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$x^2-y^2$	$\begin{bmatrix}1&0\\0&-1\end{bmatrix}$
$2xy$	$\begin{bmatrix}0&1\\1&0\end{bmatrix}$

102.13 Positive Definite Forms

Over $\mathbb{R}$ , a symmetric bilinear form $B$ is positive definite if

B(v,v)>0

for every nonzero $v\in V$ .

Positive definite symmetric bilinear forms are exactly real inner products.

They define norms by

\|v\|=\sqrt{B(v,v)}.

They also define angles through

\cos\theta = \frac{B(u,v)} {\sqrt{B(u,u)}\sqrt{B(v,v)}}.

Positive definiteness is stronger than nondegeneracy. A positive definite form is nondegenerate, but a nondegenerate form may have both positive and negative directions.

102.14 Indefinite Forms

A symmetric bilinear form is indefinite if it takes both positive and negative values on diagonal inputs.

For example, on $\mathbb{R}^2$ ,

B(u,v)=u_1v_1-u_2v_2

has

B(e_1,e_1)=1

and

B(e_2,e_2)=-1.

The associated quadratic form is

Q(x,y)=x^2-y^2.

Indefinite forms occur in relativity, hyperbolic geometry, and optimization. They define geometry, but not Euclidean distance.

102.15 Signature

For a real symmetric bilinear form, a suitable basis can diagonalize the form as

B(x,y) = x_1y_1+\cdots+x_py_p - x_{p+1}y_{p+1} -\cdots- x_{p+q}y_{p+q}.

There may also be zero directions if the form is degenerate.

The triple

(p,q,r)

is called the inertia of the form, where:

Symbol	Meaning
$p$	Number of positive squares
$q$	Number of negative squares
$r$	Number of zero directions

For nondegenerate forms, $r=0$ . The pair $(p,q)$ is called the signature.

Sylvester’s law of inertia states that these numbers are invariant under change of basis.

102.16 Symplectic Forms

A symplectic form is a nondegenerate alternating bilinear form.

The standard symplectic form on $\mathbb{R}^{2n}$ is

\omega(u,v)=u^T J v,

where

J= \begin{bmatrix} 0 & I_n\\ -I_n & 0 \end{bmatrix}.

This matrix satisfies

J^T=-J.

Symplectic forms appear in Hamiltonian mechanics. They measure oriented phase-space area rather than length.

Unlike inner products, symplectic forms satisfy

\omega(v,v)=0

for every vector $v$ .

102.17 Change of Coordinates

Let $B$ have matrix $A$ in basis $e_1,\ldots,e_n$ .

If a vector has old coordinates $x$ and new coordinates $x'$ , with

x=Px',

then

B(u,v)=x^TAy.

Substituting $x=Px'$ and $y=Py'$ , we get

B(u,v) = (x')^T P^T A P y'.

Thus the new matrix is

A'=P^TAP.

This formula explains why bilinear forms are studied under congruence transformations.

102.18 Example

Let $V=\mathbb{R}^2$ , and define

B(u,v)=2u_1v_1+u_1v_2+u_2v_1+3u_2v_2.

Then

A= \begin{bmatrix} 2 & 1\\ 1 & 3 \end{bmatrix}.

Thus

B(u,v)=u^TAv.

For

u= \begin{bmatrix} 1\\ 2 \end{bmatrix}, \qquad v= \begin{bmatrix} 3\\ -1 \end{bmatrix},

compute

Av= \begin{bmatrix} 2 & 1\\ 1 & 3 \end{bmatrix} \begin{bmatrix} 3\\ -1 \end{bmatrix} = \begin{bmatrix} 5\\ 0 \end{bmatrix}.

Therefore,

B(u,v) = \begin{bmatrix} 1 & 2 \end{bmatrix} \begin{bmatrix} 5\\ 0 \end{bmatrix} = 5.

The associated quadratic form is

Q(x,y)=2x^2+2xy+3y^2.

Since

\det A=5>0

and the leading entry is positive, this form is positive definite.

102.19 Summary

Bilinear forms are scalar-valued maps that are linear in two arguments.

Concept	Meaning
Bilinear form	Map $V\times V\to F$ , linear in each input
Matrix representation	$B(u,v)=x^TAy$
Symmetric form	$B(u,v)=B(v,u)$
Alternating form	$B(v,v)=0$
Nondegenerate form	Zero radical
Quadratic form	$Q(v)=B(v,v)$
Symplectic form	Nondegenerate alternating form

Bilinear forms make it possible to define orthogonality, length-like quantities, area-like quantities, dual-space identifications, and quadratic expressions. They are one of the main algebraic structures connecting linear algebra with geometry.