# A - B is semidefinite, what the relationship about their eigenvalues? [closed]

$A, B$ 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 $i$ ?

• Horn and Johnson, Matrix Analysis, Corollary 7.7.4. – Federico Poloni Nov 4 '13 at 16:34
• $A \ge B \implies \lambda_i(A) \ge \lambda_i(B)$. – Suvrit Nov 4 '13 at 17:43