Nearest defective matrices and the geometry of ill-conditioning
James Weldon Demmel · 1990
Abstract The problem of finding the nearest defective matrix to a given matrix has attracted the attention of J. H. Wilkinson as well as many other researchers in numerical linear algebra. In this paper we review the motivation for this probelm, and the contributions made by Wilkinson and others toward its solution. We also present a general theory of ill-conditioning which explains many of these results and relates them to analogous results about the distance to the nearest singular matrix, the distance to the nearest polynomial with a multiple root, and in fact the distance from any problem whose solution is an algebraic function to the nearest ‘ill-posed’ problem.