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Let $A$ and $B$ be symmetric non-negative matrices. If $A\geq B$ (i.e., $A-B$ is a nonnegative matrix), can we say that $\lambda_i(A) \geq \lambda_i(B)$ for all $i$, where $\lambda_i$ denotes the i--th largest eigenvalue

(I know it holds for the largest eigenvalue but I am interested in the second largest one). If it does not always hold, what condition on it makes my statement true.

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  • $\begingroup$ trivial application of minmax formulae. $\endgroup$
    – Denis Serre
    Commented Mar 22, 2013 at 21:43

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Consider $\pmatrix{1 & 1\cr 1 & 1\cr}$ and $\pmatrix{1 & 0\cr 0 & 1\cr}$. Or did you mean positive semidefinite instead of nonnegative?

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  • $\begingroup$ Thanks, your counterexample disproves my statement. Now I am looking for a condition that makes my statement true. $\endgroup$
    – Jamil Tau
    Commented Mar 23, 2013 at 12:14

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