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.