Prove that the ellipsoid $x^T W x \leq 1$ is invariant under $f (x) = A x$

Question

Given matrix $W \succ 0 $ and a set $\mathcal{Z} := \{z \mid z^T W z \leq 1\}$, prove that if $Az \in \mathcal{Z}$ and $z \in\mathcal{Z}$, then the following inequality holds

$$ A^T W A - W \preceq 0$$

Answer 1

Hint: Begin by showing that if $z$ is a vector such that $z^TWz = 1$, then we know that $$ z^T(A^TWA - W)z \leq 0 $$ Using the fact that $W$ is invertible, conclude that $z^TWz \leq 0$ for every $z$, so that $A^TWA - W$ is negative definite by definition.