III-conditioned matrices are componentwise near to singularity
For a square matrix normed to 1, the normwise distance to singularity is well known to be equal to the reciprocal of the condition number. In this paper we give an elementary and self-contained proof for the fact that an ill-conditioned matrix is also not far from a singular matrix in a componentwise sense. This is shown to be true for any weighting of the componentwise distance. In other words, for matrix inversion, "ill conditioned" means "nearly ill posed" in the normwise and also in the componentwise sense. © 1999 Society for Industrial and Applied Mathematics.
Sign-real spectral radius