Sign of the eigenvalues of the product of 3 matrices

Question

I'm trying to find the sign of the eigenvalues of a certain $n\times n$ symmetric matrix $M$. I know that i can write this matrix as $$M = PDP^T,$$

with $D$ $p\times p $ diagonal with positive or null entries, and $P$ a $n \times p$ matrix. Can I conclude that the eigenvalues of $M$ are positive or null ?

Answer 1

For all $x\in\mathbb{R}^n$,

$$\langle x, Mx\rangle=\langle x, PDP^T x\rangle=\langle P^T x, DP^T x\rangle=\langle D^{1/2} P^T x, D^{1/2} P^T x\rangle\ge 0.$$

So all eigenvalues of $M$ are nonnegative.

Answer 2

A couple of things that are useful to note. $PP^\top$ is symmetric positive semi-definite.

I think furthermore, denote $\lambda_1,d_1$ the smallest eigenvalue of $PP^\top$ and smallest entry of the main diagonal of $D$ respectively. $\lambda_2,d_2$ for largest. Then I believe $\sigma(PDP^\top)\subset[\lambda_1d_1, \lambda_2d_2]$ by Cauchy's Inequality on matrix products.

I think this gives the answer to your problem.