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Signature

The signature of a symmetric matrix $A$ is the triple $(p, z, n)$ counting how many eigenvalues are positive, zero, and negative respectively, with $p + z + n = n$ (total). For example, a $4 \times 4$ matrix with eigenvalues $\{3, 1, 0, -2\}$ has signature $(2, 1, 1)$ .

The signature completely captures the qualitative shape of the quadratic form — two matrices with the same signature define quadratic forms with the same geometric character (both are saddles, or both are bowls, etc.). This is a consequence of Sylvester's Law of Inertia: for any invertible change of coordinates $\mathbf{x} = M \mathbf{x}'$ , the signature is preserved. So the signature is an intrinsic property of the quadratic form itself, not of its matrix representation.

Formal View

Definition 8.7 — Signature

The signature of a symmetric matrix

A

with eigenvalues

\lambda_1, \ldots, \lambda_n

is the triple

(p, z, n)

where:\n

p = \#\{i : \lambda_i > 0\}

,\n

z = \#\{i : \lambda_i = 0\}

,\n

n = \#\{i : \lambda_i < 0\}

By Sylvester's Law of Inertia, the signature is preserved under congruence transformations $A \mapsto M^\top A M$ for invertible $M$ .

Why This Matters

Signature tells you the qualitative shape of a quadratic — without needing to know the exact eigenvalues.

In special relativity, the metric tensor has signature $(1, 0, 3)$ — one positive and three negative directions.
Morse theory: the index (number of negative eigenvalues) of a critical point classifies it as a minimum, saddle, or maximum.
In mechanics, the signature of the inertia tensor determines the nature of rotational motion.

Learning Resources

Sylvester's law of inertia

MIT OpenCourseWare

Discussion of the invariance of eigenvalue signs under coordinate changes.

45 min

Quadratic forms and their classification

Khan Academy

How eigenvalue signs classify quadratic forms.

12 min

Quiz

Question 1

A symmetric $5 \times 5$ matrix has eigenvalues $\{4, 2, 0, -1, -3\}$ . What is its signature?

Question 2

Two symmetric matrices with the same signature always have the same eigenvalues.

Common Mistakes

Confusing signature with the multiset of eigenvalues — signature only records sign counts, not magnitudes.
Forgetting to count multiplicity — a repeated eigenvalue contributes to the signature count multiple times.