This question is related to [Ask some matrix eigenvalue inequalities.](https://mathoverflow.net/questions/70689/ask-some-matrix-eigenvalue-inequalities)


Let $\begin{bmatrix}
A& B \\\\ B^\* &A
\end{bmatrix}$ be positive semidefinite. Is it true $\lambda\_i^{1/2}(B^\*B)\le \lambda\_i(A)$? Here, $λ\_i(⋅)$ means the ith largest eigenvalue of ⋅. 



A matrix eigenvalue problem.
This is also false. Here is a counterexample.


A = \begin{bmatrix}
 1 & 1/\sqrt{2}\\\\
 1/\sqrt{2} & 1
\end{bmatrix}


B = \begin{bmatrix}
 0 & -1/\sqrt{2}\\\\
 1/\sqrt{2} & 0
\end{bmatrix}


Then, the said block matrix has eigenvalues $(0,0,2,2)$, while


$\lambda^{1/2}(B^TB) = (1/\sqrt{2},1/\sqrt{2})$ and


$\lambda(A) = (1+1/\sqrt{2},1-1/\sqrt{2}))$


