Question about a block positive semidefinite matrix.

Question

Somebody could tell me if this result is true?

$\left( \begin{array}{cc} A & B \\ B^T & C \\ \end{array} \right)$ is positive semidefinite if and only if $\left( \begin{array}{cc} HAH^T & HB \\ B^TH^T & C \\ \end{array} \right)$ is positive semidefinite, for arbitrary H.

Answer 1

Yes. If $Y$ is a positive semidefinite real matrix and $X$ is a matrix, then $XYX^T$ is positive semidefinite (when the sizes make sense). You can apply this with $Y$ your block matrix and $X=\begin{pmatrix}H&0\\0&I\end{pmatrix}$.