We show that every $n \times n$ real matrix $A$ is within distance $\delta \|A\|$ in the operator norm of an $n\times n$ real matrix $A’$ whose eigenvectors have condition number $\tilde{O}(\text{poly}(n)/\delta)$. In fact, we show that with high probability, an additive i.i.d. sub-Gaussian perturbation of $A$ has this property. Up to log factors, this confirms a speculation of E.B. Davies.

Based on joint work with Ashwin Sah (MIT) and Mehtaab Sawhney (MIT)

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