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This question is related to 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 ⋅.

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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

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

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