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$A, B$ are two symmetric matrices, if $ A-B $ is semidefinite (i.e.$ A - B \geq 0$), if we rearrange the eigenvalues of two matrices, $\lambda_1 (A) \geq \lambda_2 (A) \geq ... \geq \lambda_n (A)$, and the same for $ B $, can we say $ \lambda_i (A) \geq \lambda_i (B) $ for each $i$ ?

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    $\begingroup$ Horn and Johnson, Matrix Analysis, Corollary 7.7.4. $\endgroup$
    – Federico Poloni
    Commented Nov 4, 2013 at 16:34
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    $\begingroup$ $A \ge B \implies \lambda_i(A) \ge \lambda_i(B)$. $\endgroup$
    – Suvrit
    Commented Nov 4, 2013 at 17:43

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