Today I found a problem concerning positive semidefinite matrix:
Suppose $A$ is a positive semidefinite matrix, with eigenvalues: $\lambda_1\geq\dots\geq\lambda_n$, and diagonal elements: $a_{11}\geq\dots\geq a_{nn}$. Now prove that, for any $k\leq n$, $\sum_{i=1}^k\lambda_i\geq\sum_{i=1}^ka_{ii}$.
Now for $k=1,n-1,n$, the problem is trivial. But what about the other $k$?
This proposition is very likely to be true as I've run some simulation on MATLAB, without finding any exception.
However this seems hard to prove. I think maybe some transformation needs to be done. Anyone has any clue? Thanks~
Besides there are some background in PCA. I shall provide some supplement if necessary later.
By Sylvester's criterion, a symmetric matrix is positive semi-definite if and only if all its principal minors are non-negative. This in particular means that all the principal submatrices are also positive semi-definite themselves. You checked the result for $k=1$, so it is true in general by induction.