# Chapter 102. Bilinear Forms

# 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

$$
B(u,v)=u_1v_2-u_2v_1.
$$

If

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

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

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

If

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

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

$$
\Phi_B : V \to V^*
$$

by

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

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.