Estimates on the condition number of random rank-deficient matrices

Let r < m < n is an element of N and let A be a rank r matrix of size m x n, with entries in K = C or K = R. The generalized condition number of A, which measures the sensitivity of Ker(A) to small perturbations of A, is defined as kappa(A) = vertical bar A vertical bar A(dagger)vertical bar, where (dagger) 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 x n matrices is endowed with the natural probability measure coming from the Gaussian measure in K-m (x n). We also prove an upper-bound estimate for the expected value of log kappa in this setting.