Mathematics Colloquia and Seminars

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I will survey a line of work showing that a small i.i.d. perturbation of any matrix behaves more or less like a normal matrix, in the sense that its eigenvectors are well-conditioned with high probability. This was conjectured by E.B. Davies and is useful in certain numerical algorithms. Interestingly, the proofs are quite different for real and complex perturbations. I will also mention some open problems that remain, with motivation from both pure and applied mathematics. No background is assumed.