I know positive semi-definite matrices are generalizations of non-negative numbers. So "ordering" of the two systems should be pretty much like each other. How to prove the following theorem?
For two symmetric $X$ and $Y$, if $X \geq Y$, then $\lambda_i(X) \geq \lambda_i(Y)$, for every $i$. $\lambda_i(\cdot)$ denotes the $i$-th largest eigenvalue.
And what about the converse statement? Is it true?
Thanks a lot.