Abstract: Let r ? m ? n ? ? and let A be a rank r matrix of size m × n, with entries in ?? = ? or ?? = ?. The generalized condition number of A, which measures the sensitivity of Ker(A) to small perturbations of A, is defined as ?(A) = |A?A?|, where ? denotes Moore?Penrose pseudoinversion. In this paper we prove sharp lower and upper bounds on the probability distribution of this condition number, when the set of rank r, m × n matrices is endowed with the natural probability measure coming from the Gaussian measure in ??m× n. We also prove an upper-bound estimate for the expected value of log ? in this setting.

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