Ill-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 selfcontained 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 words: Ill-conditioned means for matrix inversion nearly ill-posed also in the componentwise sense.