---
tag: 0KQZYGEJTF
kind: definition
name: Signature of a quadratic form
msc: [15A63, "11E04"]
defines: 07EZA80NEH
uses: [0HZNE5QZDJ, 0WFED3K3DQ]
sources:
  - url: https://en.wikipedia.org/wiki/Metric_signature
    locator: Definition
---

## Definition

Let $V$ be a real vector space of finite dimension $n$, and let $Q$ be a [[quadratic form|0RBQS1JJCR]] on $V$.
Choose a basis in which $Q(x_1 e_1 + \cdots + x_n e_n) = a_1 x_1^2 + \cdots + a_n x_n^2$.
Let $p$, $q$ and $r$ be the numbers of indices $i$ with $a_i > 0$, $a_i < 0$ and $a_i = 0$.
The *signature* of $Q$ is the triple $(p, q, r)$.

## Remarks

By [[Sylvester's law of inertia|0WFED3K3DQ]], the triple does not depend on the choice of the basis.
For a symmetric matrix $A$ of $Q$, the numbers $p$, $q$ and $r$ count the positive, negative and zero eigenvalues of $A$, with multiplicity.
Conventions vary: some authors write only $(p, q)$, or the number $p - q$.
The Minkowski form $x_1^2 - x_2^2 - x_3^2 - x_4^2$ on $\R^4$ has $p = 1$, $q = 3$ and $r = 0$.

## Explanations

### L1 human

The signature counts the plus signs, the minus signs and the zeros in a real quadratic form, after diagonalisation.
It is the complete invariant of a real quadratic form.
Two real quadratic forms are equivalent exactly when they have the same signature.