Prove that if $M$ is a symetric positive definite matrix then $(S^T)MS$ is also symetric positive definite

Question

I'm asked to prove that with $S$ being any non singular matrix , if $M$ is a symetric positive definite matrix then $S^TMS$ is also symetric positive definite.

Answer 1

Note that since $S$ is invertible, $S$ is a bijeciton on the vector space.

$x \ne 0 \iff Sx \ne 0$

$M$ is positive definite, so $x^T M x = \langle Mx,x\rangle \ge 0$, and $x^T M x = 0 \iff x = 0$.

$$\langle S^T M Sx,x\rangle = x^T S^T M S x = (Sx)^T M Sx = \langle MSx,Sx\rangle \ge 0$$ and $$\langle MSx,Sx\rangle = 0 \iff Sx = 0 \iff x = 0.$$