A diagonalizable matrix has linearly independent
eigenvectors. Since the set of nondiagonalizable
matrices has measure zero, every matrix is the
limit of diagonalizable matrices. We prove a
quantitative version of this fact: every n x n
complex matrix is within distance delta of a matrix
whose eigenvectors have condition number poly(n)/delta,
confirming a conjecture of E. B. Davies. The proof
is based on regularizing the pseudospectrum with a
complex Gaussian perturbation.
Joint work with J. Banks, A. Kulkarni, S. Mukherjee.