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(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.

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  • $\begingroup$ Perhaps $\|\cdot\|_2$ should just be $\|\cdot\|$ in statement 1? $\endgroup$
    – Chris Ramsey
    Commented Mar 14, 2016 at 12:44
  • $\begingroup$ I have double-checked, it's $\|\cdot\|_2$ $\endgroup$
    – Drn004
    Commented Mar 14, 2016 at 12:49
  • $\begingroup$ This looks a lot like homework. Voting to close. $\endgroup$
    – Michael Renardy
    Commented Mar 14, 2016 at 14:07
  • $\begingroup$ It can be closed, Denis Serre's answer has been more than helpful. $\endgroup$
    – Drn004
    Commented Mar 14, 2016 at 14:08

1 Answer 1

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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).

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