Proving that every symmetric matrix is congruent to the following canonical form

Question

Let $B\bf $ be a real symmetric matrix of order $n\times n$. Then show that there exists an invertible matrix $\bf P$ such that $\bf P'AP=\begin{pmatrix}\bf I & \bf0 & \bf0 \\\bf0 & \bf-I & \bf0 \\\bf0 & \bf0 & \bf0* \\\end{pmatrix}$; $\bf I$ and $\bf-I$ denote the number of 1s and -1s in the diagonal and $\bf0*$ denotes the number of 0s.

I seriously have no idea now to begin this proof and cannot find any resources online which state the proof.

Can anyone help me out or redirect me to the proof?

Answer 1

This is a basic result called Sylvester's law of inertia. You can find the proof in any good book on Linear Algebra. See for instance

J. J. Rotman, Advanced Modern Algebra (second edition), Theorem 8.84 p. 692.