I saw this theorem stated in a paper without proof and I have difficulty proving it.
If $A$ is an $n\times n$ matrix with non-negative off-diagonal entries, let $s(A)$ be the real eigenvalue such that $s(A)> Re(\lambda(A))$ for all eigenvalues $\lambda$ of $A$. From properties of $Z$-matrices, we know such an eigenvalue exists.
Let $M$ and $L$ are two matrices with non-negative off-diagonal entries and $M<L$ (implying $m_{ij}<l_{ij}$ for all $i,j$) , how do we prove $s(M)<s(L)$?