Canonical Forms and Sylvester's Law

Every symmetric bilinear form can be reduced to a canonical diagonal form. Sylvester's law of inertia states that the signature is a complete invariant over the reals.

TheoremSylvester's Law of Inertia

Let $Q(\mathbf{x}) = \mathbf{x}^TA\mathbf{x}$ be a real quadratic form. There exists invertible matrix $P$ such that: $P^TAP = \begin{bmatrix} I_p & 0 & 0 \\ 0 & -I_q & 0 \\ 0 & 0 & 0 \end{bmatrix}$

where $I_p$ is the $p \times p$ identity. The numbers $(p, q)$ are uniquely determined by $A$ and called the signature. Any two quadratic forms with same signature are equivalent under congruence.

This canonical form reveals the intrinsic geometric character: $p$ positive squares, $q$ negative squares, and $n-p-q$ zeros.

ExampleReducing to Canonical Form

For $Q(\mathbf{x}) = x_1^2 + 2x_1x_2 + 2x_2^2$ :

Complete the square: $(x_1 + x_2)^2 + x_2^2$

Set $y_1 = x_1 + x_2, y_2 = x_2$ : $Q = y_1^2 + y_2^2$

Signature: $(2, 0, 0)$ (positive definite).

DefinitionCongruence of Matrices

Matrices $A$ and $B$ are congruent if $B = P^TAP$ for some invertible $P$ .

Congruence preserves symmetric/skew-symmetric character and determines equivalence of bilinear forms. Unlike similarity (which preserves eigenvalues), congruence preserves signature.

TheoremClassification over Different Fields

Over $\mathbb{R}$ : Symmetric bilinear forms classified by signature $(p, q, r)$ (Sylvester)

Over $\mathbb{C}$ : All non-degenerate forms equivalent; classified only by rank (complex numbers have no ordering)

Over finite fields: Classification depends on field structure; more subtle invariants needed

ExampleIndefinite Forms

$Q(\mathbf{x}) = x_1^2 - x_2^2$ has signature $(1, 1, 0)$ . The level set $Q = 0$ gives $x_1^2 = x_2^2$ , a pair of lines (degenerate hyperbola). Non-zero level sets $Q = c$ give hyperbolas.

In spacetime physics, the Minkowski metric $ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$ has signature $(3, 1, 0)$ , distinguishing timelike and spacelike directions.

Remark

Sylvester's law is fundamental because it provides a complete classification: two real symmetric matrices represent "the same" quadratic form (up to change of coordinates) if and only if they have the same signature. This result extends the spectral theorem's philosophy—canonical forms reveal intrinsic structure—to the congruence setting.