Condition numbers of rectangular systems and bounds for generalized inverses

A natural extension of the notion of condition number of a matrix to the class of all finite matrices is shown to enjoy properties similar to the classical condition number. For example, the relative distance to the set of all matrices of smaller rank is just the reciprocal of this generalized condition number. The question of whether a matrix with a small generalized condition number must also have a generalized inverse of small norm is then studied. The answer turns out to be norm dependent. In particular, only if p is 1 or 2 must an intrinsically well-conditioned full rank matrix in the l_p sense have a nicely bounded generalized inverse; in particular, in the l_∞ norm this need not be true. These facts are consequences of recent results in Banach space theory.