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

Abstract

Let L be the in nite lower triangular Toeplitz matrix with rst column (μ, a1, a2, ..., ap, a1, ..., ap, ...)T and let D be the in nite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. B¨unger and Rump have shown that if p = 2 and certain linear inequalities between the parameters μ, a1, a2, are satis ed, then the singular values of any nite 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 de ned 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.