\begin{definition}[Definition:Signature of Symmetric Bilinear Form]
Let $\struct {V, q}$ be the scalar product space.
Let $V^*$ be the vector space dual to $V$.
Suppose there exists a basis $\tuple {\beta^i}$ for $V^*$ such that $q$ is expressible as:
:$q = \paren {\beta^1}^2 + \ldots + \paren {\beta^r}^2 - \paren {\beta^{r + 1}}^2 - \ldots - \paren {\beta^{r + s}}^2$
where:
:$r, s \in \N : r + s = n$.
Then the ordered pair $\tuple {r, s}$ is known as the '''signature of $q$'''.
\end{definition}