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

• Autor(a) principal: Solary, Maryam Shams
• Data de Publicação: 2021
• Outros Autores: Kovacec, Alexander, Capizzano, Stefano Serra
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10316/105479; https://doi.org/10.1515/spma-2020-0127
• topic: Toeplitz related matrix; triangular matrix; singular value; in nite-dimensional matrix; asymptotics of linear recurrences

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.