Bilinear form/Symmetric/Type/Inertia law/Fact

Sylvester's law of inertia

Let ${}V$ be a finite-dimensional real vector space, endowed with a symmetric bilinear form ${}\left\langle -,-\right\rangle$ of type ${}(p,q)$.

Then the

Gram matrix of ${}\left\langle -,-\right\rangle$ with respect to any orthogonal basis is a diagonal matrix

with ${}p$ positive and ${}q$ negative entries.

Proof, Write another proof