Bound on the eigenvalues of PSD matrix Given That A and B are two PSD (positive semi-definite) real matrices and the following holds
$$
A \leq B 
$$
(meaning that $$ B-A $$ is also PSD) 
can I bound the eigenvalues of A using the eigenvalues of B?
Thanks

We know that the trace of a matrix is equal to the sum of its eigenvalues.  Further, since each entry in $A$ is no greater than the corresponding entry of $B$, it follows that tr($A)\le$tr($B)$. Thus, the sum of the eigenvalues of $A$ can be no greater than the sum of the eigenvalues of $B$, in other words, you could say 1) That, on average, the eigenvalues of $A$ must be less than or equal to the eigenvalues of $B$; 2)That the range of possible eigenvalues for $A$ are bounded above by tr$(B)$.
