(I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty)


How can I prove that the following statements are equivalent?


1. $\lambda$ is an eigenvalue of $A+\delta A$, where $\|\delta A\|\_{2}\leq \epsilon$
2. $\exists u\in \mathbb{C}^{m}$ such that $\|(A-\lambda I)u\|\_{2}\leq\epsilon$ and $\|u\|\_{2}=1$
3. $\sigma\_{n}(\lambda I - A)\leq \epsilon$, where $\sigma\_{n}$ is the smallest singular value of A
4. $\|(\lambda A - I)^{-1}\|\geq \epsilon^{-1}$


I am using An Introduction to Numerical Analysis by Endre Süli and David F. Mayers but it's not been very helpful. If you could recommend me another textbook I would be very grateful.



Eigenvalue-related statements
This is rather classical. The set of such $\lambda$'s is called the $\epsilon$-pseudo-spectrum. It is presented in many books on numerical linear algebra. I suggest S.K. Godunov *Modern aspects of linear algebra*, AMS (1998).


