A diagonalizable matrix has linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every matrix is a limit of diagonalizable matrices. We prove a quantitative version of this fact: every n x n complex matrix is within distance delta in the operator norm 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 of an arbitrary matrix with a complex Gaussian perturbation. Joint work with J. Banks, A. Kulkarni, S. Mukherjee.