<s>[INST] Inverse of $A^T Y A$ Is there a formula for the inverse of a matrix
$$ X = A^\top Y A $$
in terms of $A$ and $Y$, given that $A \in F^{m \times n}$ is full rank with $m >n$, and $Y$ is positive definite?
 [/INST] Even with the restriction $m>n$, it need not be the case that $X=A^\top Y A$ be invertible. As a counterexample, suppose $m=2,n=1$. Then the choice $$Y=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},\; A=\begin{pmatrix} 1 \\ 1\end{pmatrix}$$
yields $$X = A^\top Y A = \begin{pmatrix} 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix}=\begin{pmatrix} 1 & -1 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix}=0$$ which is plainly not invertible. 
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