I recently had an interesting observation but fail to come up with a rigorous explanation. What I observed is stated as follows. I am wondering if anyone is familiar with it.
Let $A$ be a real matrix, $\lambda_{\text{max}}$ be the largest real part of its eigenvalues. If we add (resp., substract) real positive numbers from its $i$-th diagonal entry, the new $\lambda_{\text{max}}$ will increase (resp., decrease).
For example, when $A=\left[ -100 -1000 ; 1000\quad 40\right]$, $\lambda_{\text{max}}=-30$, but when we change $A(2,2)$ to 10, $\lambda_{\text{max}}$ decreases to -40.
I have been thinking about it for many days, I really hope that someone could help me with it. Thank you in advance!
I'll try to point you to https://en.wikipedia.org/wiki/Gershgorin_circle_theorem. To me it seems exactly what you are looking for.