On minimal singular values of random matrices with correlated entries

Let X be a random matrix whose pairs of entries X-jk and X-kj are correlated and vectors (X-jk, X-kj), for 1 <= j < k <= n, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that EXjk = 0, EX (2)(jk) = 1, for any j, k = 1,..., n and EXjkXkj = rho for 1 <= j < k = n. Let M-n be a non-random nxn matrix with parallel to M-n parallel to <= K-n(Q), for some positive constants K > 0 and Q >= 0. Let s(n) (X + M-n) denote the least singular value of the matrix X + M-n. It is shown that there exist positive constants A and B depending on K, Q, rho only such that P(sn(X + 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.