Lower Bounds for the Smallest Singular Value of Certain Toeplitz-like Triangular Matrices with Linearly Increasing Diagonal Entries

摘要

Let L be a lower triangular n x n-Toeplitz matrix with first column (mu, a, beta, a, beta,...) T, where mu, a, beta = 0 fulfill a-beta. [0, 1) and a. [1, mu+ 3]. Furthermore let D be the diagonal matrix with diagonal entries 1, 2,..., n. We prove that the smallest singular value of the matrix A := L + D is bounded from below by a constant. =.(mu, a, beta) > 0 which is independent of the dimension n.