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This question is related to http://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 ⋅. |
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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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