Bilinear form/Symmetric/Eigenvalue criterion/Fact

Eigenvalue criterion for the type

Let ${}\left\langle -,-\right\rangle$ be a symmetric bilinear form on a finite-dimensional real vector space, and let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$. Let ${}G$ denote the Gram matrix of ${}\left\langle -,-\right\rangle$ with respect to this basis.

Then the

type ${}(p,q)$ of the form has the following interpretation: ${}p$ is the sum of the dimensions of the eigenspaces of ${}G$ for positive eigenvalues,

and ${}q$ is the sum of the dimensions of the eigenspaces of ${}G$ for negative eigenvalues.

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