Date(s) - 17/01/2020
16 h 00 min - 17 h 00 min
FRUMAM, St Charles (2ème étage)
Nikhil SRIVASTAVA (UC Berkeley)
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 adding a complex Gaussian perturbation to the matrix and studying its pseudospectrum.
Finally, we mention a recent application of this result to numerical analysis, yielding the fastest known provable algorithm for diagonalizing an arbitrary matrix.
Joint work with J. Banks, A. Kulkarni, S. Mukherjee, J. Garza Vargas.