Can we say any matrix whose quadratic form is $xy + z^2$ can be diagonalised to a Matrix who has two positive and one negative Eigen value?

Question

Can we say any matrix whose quadratic form is $xy + z^2$ can be diagonalised to a Matrix who has two positive and one negative Eigen value?

If it can , from which theorem you can. I have been having a really bad time to understand one problem.

I would be highly obliged if anyone help me to get my doubt cleared.

Thank You in advance.

Answer 1

Yes it is possible by Sylvester's law of inertia since the matrix associated to $xy+z^2$ is

$$B=\begin{bmatrix}0&\frac12&0&0\\\frac12&0&0&0\\0&0&1&0\\0&0&0&0 \end{bmatrix}$$

with signature

$$n_+=2,n_-=1,n_0=1$$