# Is this matrix product positive semidefinite?

Let $A,B\in\mathbb{C}^{p\times p}$ s.t. $A$ and $B$ are contractions and $I_p-AA^*$ and $BB^*$ both are positive semidefinite. I want to show that the matrix $$B(I_p-\sqrt{I_p-AA^*}\sqrt{I_p-AA^*}^*)B^*$$ is positive semidefinite. Are the conditions enough to show this? I'd much appreciate any help!

• Note that $\sqrt{I_p-AA^*}$ is itself Hermitian and positive definite. So, your matrix is simply $$B(I - (I - AA^*))B^* = B(AA^*)B^*$$ And yes; this matrix is necessarily positive semidefinite. – Omnomnomnom Jun 3 '18 at 20:04
• Of course! That resolves it. – Vorhang Jun 3 '18 at 20:11