Vishesh Jain, Stanford University, On the real Davies’ conjecture
ZoomWe 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…