So $X$ and $Y$ are Hermitian matrices (or just symmetric real) of size $n$ by $n$ and suppose $Y\succeq X$, namely $Y-X$ is positive-semidefinite. Now write the eigenvalues of $Y$ as $\alpha_1\leq\ldots\leq \alpha_n$, and the eigenvalues of $X$ as $\beta_1\leq\ldots\leq \beta_n$. Is is necessarily true that $\alpha_i\geq\beta_i$ for all $i$?
I might be able to solve this myself (although with time I am less sure), but it should be be much easier for whoever already knows the answer. A quick reference would do, thanks.