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arxiv: 0802.3956 · v4 · submitted 2008-02-27 · 🧮 math.PR · math.FA

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The smallest singular value of a random rectangular matrix

Mark Rudelson , Roman Vershynin
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classification 🧮 math.PR math.FA
keywords matrixsingularsmallestvalueestimateprobabilityrandomsqrt
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We prove an optimal estimate on the smallest singular value of a random subgaussian matrix, valid for all fixed dimensions. For an N by n matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order \sqrt{N} - \sqrt{n-1} with high probability. A sharp estimate on the probability is also obtained.

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