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.