Every matrix is congruent to a diagonal matrix

Question

I think I've read somewhere that any square matrix $M$ can be decomposed as $M=P^TDP$, where $D$ is a diagonal matrix. Such statement looks very much like the spectral theorem, although in this case $P$ is not orthogonal in general (thus such fact, even if it is true, would be of little use). Does anyone know if my recollection is correct and, if so, a reference for that theorem. Thanks!

Answer 1

This can't be true in general because $P^TDP$ is always symmetric and $M$ isn't necessarily.