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

Abstract

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