Let $A$ be a real matrix, if some of its diagonal entries perturbed by subtracting postive real numbers, then what can we say about the eigenvalues of perturbed matrix in comparison to eigenvalues of the original?
I know that if every diagonal entry of $A$ decreased uniformly by $k(>0)$ amount then real part of every eigenvalues of perturbed will be shifted by $k$ amount left side.
I am especially intrested in any comparison between $\max Re\lambda (A)$ and $ \max Re\lambda (A -D),~D$ is a diagonal matrix whose at least one entry is positive.