Estimates on the condition number of random rank-deficient matrices

Let r ≤ m ≤ n ∈ 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 κ (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 x n matrices is endowed with the natural probability measure coming from the Gaussian measure in K m×n . We also prove an upper-bound estimate for the expected value of log κ in this setting.