Question about a block positive semidefinite matrix.

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.

• One direction is obvious from taking $H=I$. I'm curious, what inspired the question? Also could you please make explicit that you are talking about matrices with real entries? Do you know that if $A$ is a positive semidefinite real matrix and $X$ is a real matrix, then $XAX^T$ is positive semidefinite? – Jonas Meyer Mar 16 '17 at 5:51
• I'm working with covariances matrices, and right now, I have a matrix in this form $\left( \begin{array}{cc} HAH^T & HB \\ B^TH^T & C \\ \end{array} \right)$. I create some numerical experiments in MATLAB, and it looks like is true, but I cannot prove it. I tried to find some references, but nothing. – Juan Pablo Soto Quirós Mar 16 '17 at 5:53
• Yes, I know that. In fact, my issue is to prove that $C-B^TH^T(HAH^T)^\dagger HB$ is positive semidefinite. – Juan Pablo Soto Quirós Mar 16 '17 at 5:57

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 $X=\begin{pmatrix}H&0\\0&I\end{pmatrix}$.