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 (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 Km x n. We also prove an upper-bound estimate for the expected value of log in this setting.}

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Original source : http://imajna.oxfordjournals.org/cgi/content/short...