Is $X^TAX$ p.s.d if $A$ is p.s.d?

Question

Is $X^TAX$ positive semi-definite if $A$ is positive semi-definite? All matrices are real-valued. I am interested in both cases when i) $X$ is rectangular matrix ii) $X$ is square matrix.

Answer 1

If $A$ is $n\times n$ and $X$ is $n\times m$, then for all $v\in\mathbb{R}^m$ we have $$ v^TX^TAXv=(Xv)^TA(Xv)\geq 0$$ since $A$ is assumed to be positive semidefinite. Therefore $X^TAX$ is positive semidefinite as well.