On minimal singular values of random matrices with correlated entries

2013/09/23 by Götze, Friedrich, Naumov, Alexey, Tikhomirov, Alexander · 2 citations
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1309.5711

Abstract

Let \mathbf X be a random matrix whose pairs of entries X_jk and X_kj are correlated and vectors (X_jk,X_kj), for 1≤ j0 and Q≥ 0. Let s_n(\mathbf X+\mathbf M_n) denote the least singular value of the matrix \mathbf X+\mathbf M_n. It is shown that there exist positive constants A and B depending on K,Q,ρ only such that ℙ(s_n(\mathbf X+\mathbf M_n)≤ n^-A)≤ n^-B. As an application of this result we prove the elliptic law for this class of matrices with non identically distributed correlated entries.

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