Signature of a quadratic form

Definition

Let $V$ be a real vector space of finite dimension $n$, and let $Q$ be a quadratic form 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, 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.