Classification Theorems

Classification theorems for bilinear and quadratic forms provide complete invariants determining when two forms are equivalent. The answer depends crucially on the underlying field.

TheoremClassification over $\mathbb{R}$

Two real symmetric bilinear forms (or quadratic forms) are congruent if and only if they have the same signature $(p, q, r)$ .

This means every real quadratic form is congruent to exactly one of: $\sum_{i=1}^p y_i^2 - \sum_{i=p+1}^{p+q} y_i^2$

The rank is $p + q$ , and signature completely classifies the form.

TheoremClassification over $\mathbb{C}$

Over complex numbers, symmetric bilinear forms are classified only by rank. Every non-degenerate form of rank $r$ is congruent to: $y_1^2 + y_2^2 + \cdots + y_r^2$

The distinction between positive/negative disappears since $-1 = i^2$ is a square in $\mathbb{C}$ .

TheoremWitt's Cancellation Theorem

If $V_1 \oplus W \cong V_2 \oplus W$ as spaces with bilinear forms (where $W$ is non-degenerate), then $V_1 \cong V_2$ .

This allows "canceling" common summands when comparing forms, analogous to cancellation in arithmetic.

ExampleConic Section Classification

In $\mathbb{R}^2$ , quadratic forms $Q(\mathbf{x}) = ax_1^2 + 2bx_1x_2 + cx_2^2$ classify conics via discriminant $\Delta = b^2 - ac$ :

$\Delta < 0$ : Ellipse (signature $(2,0)$ or $(0,2)$ )
$\Delta > 0$ : Hyperbola (signature $(1,1)$ )
$\Delta = 0$ : Parabola (degenerate, rank 1)

The signature provides geometric classification.

TheoremSymplectic Forms

A skew-symmetric bilinear form $\omega$ is called symplectic if non-degenerate. Every symplectic form on $\mathbb{R}^{2n}$ is congruent to: $\omega_0 = \sum_{i=1}^n (x_idy_i - y_idx_i)$

This canonical form underlies Hamiltonian mechanics and symplectic geometry. Dimension must be even; symplectic forms exist only on even-dimensional spaces.

Remark

Classification theorems transform infinite families of matrices into finite lists of canonical forms. Over $\mathbb{R}$ , geometry matters: signature distinguishes elliptic from hyperbolic behavior. Over $\mathbb{C}$ , only rank survives. These results extend to more general settings (p-adic fields, function fields) with increasingly subtle invariants, forming the foundation of algebraic K-theory and quadratic form theory.