$S^TAS$ is positive semidefinite

Question

I need to prove this: If A is an n$\times$n positive semidefinite matrix, and S any $n \times m$ matrix, then $S^TAS$ is positive semidefinite.

Answer 1

$x^TS^TASx=(Sx)^TA(Sx)$. Therefore, this quantity is $\ge 0$ by hypothesis on $A$.