The smallest singular value of certain Toeplitz-related parametric triangular matrices

Let L be the infinite lower triangular Toeplitz matrix with first column (μ, a1, a2, ⋯, ap, a1, ⋯, ap, ⋯)T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, ⋯ Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters μ, a1, a2, are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for μ, a1, ⋯, ap. It depends on the asymptotics in μ of the l2-norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.