Almost Sharp Bounds for the Componentwise Distance to the Nearest Singular Matrix
Siegfried M. Rump · Linear and Multilinear Algebra · 1997
The normwise distance of a regular matrix A∊Mn to the nearest singular matrix is well known to be A -1 −1.Such a normwise distance neglects small entries in the matrix, and it does not allow for weights in a perturbation. The reciprocal A -1·E -1 of the Bauer-Skeel condition number is known to be a lower bound for the componentwise distance of A to the nearest singular matrix weighted by the nonnegative matrix E. In this paper we derive an upper bound for this componentwise distance involving the Bauer-Skeel condition number. We show that this upper bound is sharp up to a constant factor less than ,independent of A and E. For finite values of n, improved constants are given as well.