How to estimate the differences between the eigenspectrum of a complex matrix $M$ and pertubed matrix $M+E$?

Question

I was reading this text about the problem: https://arxiv.org/pdf/2006.01837

To be specific, I found this theorem on P14 The "spectral shift" theorem, that I cannot find elsewhere

I do not understand the "max" and "min" in the theorem. Can anyone provide some advice?

Thanks in advance.

Answer 1

It's another way of saying:

For every eigenvalue $\lambda'_j$ of $M+E$, there is an eigenvalue $\lambda_{j'}$ of $M$ such that $$|\lambda_j' - \lambda_{j'}| \le \operatorname{cond}(V)\|E\|.$$

We can replace "there is an eigenvalue $\lambda_{j'}$" by a min over all possible $j'$ to turn this into

For every eigenvalue $\lambda'_j$ of $M+E$, $$\min_{j'} |\lambda_j' - \lambda_{j'}| \le \operatorname{cond}(V)\|E\|.$$

Then, since this inequality holds for every $j$, it holds for the $j$ that maximizes $\min_{j'} |\lambda_j' - \lambda_{j'}|$, which gives you the expression you saw.