generalized ordering of positive semi-definite matrix by eigenvalues 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. 

I checked Roger A. Horn's matrix analysis book. It provides a very thorough analysis of the ordering defined on positive semi-definite matrices.
Here is a logic stream how to prove the original statement:
Courant-Fischer Theorem $\Rightarrow$ Weyl Theorem $\Rightarrow$ Monotonicity Theorem
The original statement is a direct consequence of Monotonicity Theorem.
And the converse statement is not true. Here is a simple example:
$A = \bigl(\begin{smallmatrix}
2&0\\ 0&4
\end{smallmatrix} \bigr)$
$B = \bigl(\begin{smallmatrix}
3&0\\ 0&1
\end{smallmatrix} \bigr)$
