$A, B$ 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 $i$ ?

Matrix Analysis, Corollary 7.7.4. $\endgroup$ – Federico Poloni Nov 4 '13 at 16:34