Bounds for the Componentwise Distance to the Nearest Singular Matrix

The normwise distance of a matrix A to the nearest singular matrix is well known to be equal to ‖A‖/cond(A) for norms being subordinate to a vector norm. However, there is no hope to find a similar formula or even a simple algorithm for computing the componentwise distance to the nearest singular matrix for general matrices. This is because Rohn and Poljak [7] showed that this is an NP -hard problem. Denote the minimum Bauer-Skeel condition number achievable by column scaling by κ. Demmel [3] showed that κ−1 is a lower bound for the componentwise distance to the nearest singular matrix. In this paper we prove that 2.4 · n1.7 · κ−1 is an upper bound. This extends and proves a conjecture by N. J. Higham and J. Demmel. We give an explicit set of examples showing that an upper bound cannot be better than n · κ−1. Asymptotically, we show that n1+ln 2+ε · κ−1 is a valid upper bound.