Given $P$ a real symmetric positive semidefinite matrix, what matrix $A$ ensure that $A^TPA$ is PSD?

Question

Suppose $P$ is a real symmetric positive semi-definite matrix

$$\forall x \in \mathbb{R}^n, x^TPx \geq 0$$

Question, for what matrices $A$ is $A^TPA$ positive semidefinite?

Obviously $I$ works, $\alpha I$ works for any $\alpha \geq 0$. Are there general classes of matrices $A$ such that this "composition" is still PSD?

Answer 1

$\forall A \in \mathbb{R}^{n \times m}$, $\forall y \in \mathbb{R}^{m \times 1}$, since $$y^TA^TPAy=(Ay)^TP(Ay) \geq 0$$

Hence, $A^TPA$ is PSD.