We prove an optimal estimate of the smallest singular value of a random sub‐Gaussian matrix, valid for all dimensions. For an N × n matrix A with independent and identically distributed sub‐Gaussian entries, the smallest singular value of A is at least of the order √ N − √ n − 1 with high probability. A sharp estimate on the probability is also obtained.
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Rudelson et al. (2009) studied this question.
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