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Quantitative Diagonalizability

Program in Applied Mathematics Colloquium

Quantitative Diagonalizability
Series: Program in Applied Mathematics Colloquium
Location: MATH 501
Presenter: Nikhil Srivastava, Mathematics Department, UC Berkley

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.