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 $\|A\|/{\rm cond}(A)$ for norms subordinate to a vector norm. However, there is no hope for finding a similar formula or even a simple algorithm for computing the componentwise distance to the nearest singular matrix for general matrices. This is because Poljak and Rohn [Math. Control Signals Systems, 6 (1993), pp. 1--9] showed that this is an NP-hard problem. Denote the minimum Bauer--Skeel condition number achievable by column scaling by $\kappa$. Demmel [SIAM J. Matrix Anal. Appl., 13 (1992), pp. 10--19] showed that $\kappa^{-1}$ is a lower bound for the com- ponentwise distance to the nearest singular matrix. In our paper we prove that $2.4 \cdot n^{1.7} \cdot \kappa^{-1}$ is an upper bound. This extends and proves a conjecture by Demmel and Higham (in the cited paper by Demmel). We give an explicit set of examples showing that such an upper bound cannot be better than $n \cdot \kappa^{-1}$. Asymptotically, we show that $n^{1+ \ln 2+\varepsilon} \cdot \kappa^{-1}$ is a valid upper bound.