Let $A \in \mathbb{C}^{n \times n}$ be some (fixed) matrix with eigenvalues $\lambda_{1},\ldots,\lambda_{n}$. Let $E$ be some random, small-normed, perturbation such that $\tilde{A} = A+E$ has eigenvalues $\tilde{\lambda}_1,\ldots,\tilde{\lambda}_{n}$.

Denote $\Delta = \min_{i \neq j}|\tilde{\lambda}_i - \tilde{\lambda}_j|$. Nguyen, Tao and Vu proved that, roughly, if $A$ is Hermitian and $E$ is a random Wigner matrix then for every fixed $a$ there exists $b = b(\|E\|)$ such that $\Delta \ge n^{-b}$ with probability at least $1-n^{-a}$.

From what I understand, no similar result concerning non-Hermitian matrices is known. Note that the specific distribution $E$ is drawn from is not important to me (as long as it is "standard").

I am looking for a slightly weaker result concerning the size of the Jordan blocks of the perturbed matrix. Namely, let $B$ be the size of the largest Jordan block of a Jordan decomposition of $A$. What is w.h.p. the size of the largest Jordan block of $\tilde{A}$?

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    $\begingroup$ Any sensible random matrix has simple eigenvalues, no? $\endgroup$ – Mariano Suárez-Álvarez Oct 20 '16 at 3:55
  • $\begingroup$ I think so too, as the set of matrices with $n$ different eigenvalues is open and dense. $\endgroup$ – user1688 Oct 20 '16 at 5:24
  • $\begingroup$ Right, but is there a specific distribution known (with non-fixed entries mean) for which such a quantitive reault be given? From what I gathered, it is not the case. Thanks. $\endgroup$ – Daniel86 Oct 20 '16 at 9:30
  • $\begingroup$ @MarianoSuárez-Álvarez: Not necessarily. Do you believe that discrete random variables are sensible? The intuition from continuous distributions suggests that the answer to the question must be 1, but proving that for, say, a matrix with independent Bernoulli entries may be difficult. $\endgroup$ – Mark Meckes Oct 20 '16 at 13:19
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    $\begingroup$ As @Mark says, the answer should be 1, but this hasn't been proved. As an application of their four moment theorem, Tao and Vu deduce a bound $O(n^{1-c})$ for some constant $c>0$ on the maximum multiplicity for a random matrix with iid entries matching the real or complex Gaussian to four moments -- see Corollary 18 here: arxiv.org/pdf/1206.1893v6.pdf. I'm not aware of any other work on eigenvalue multiplicity for non-Hermitian random matrices. $\endgroup$ – Nick Cook Oct 20 '16 at 21:32

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