Does every matrix of the form $P J_p P^\top$ have exactly $p$ positive eigenvalues?

Question

Let $J_{p}$ be the $n \times n$ diagonal matrix with $p$ ones followed by $n-p$ minus ones on its diagonal. Let $P$ be an invertible $n \times n$ real matrix (not necessarily symmetric).

How many positive eigenvalues can $P J_p P^\top$ have? Is it always equal to $p$?

I have not been able to find a counterexample (I generated a bunch of random matrices $P$) or a proof.

Answer 1

If you think of $J_p$ as the matrix of a quadratic form, $PJ_pP^{T}$ is the matrix of the same quadratic form in a different basis (the base of the columns vectors of $P$). By the law of inertia, the number of positive, negative and zero eigenvalues is preserved (see Sylvester's law of inertia). So your conjecture is correct.