Explicit inverse of a tridiagonal (p,r)-Toeplitz matrix
We have named tridiagonal (p,r)–Toeplitz matrix to those tridiagonal matrices in which each diagonal is a quasi–periodic sequence, d(p+j)=rd(j), so with period p¿N but multiplied by a real number r. We present here the necessary and sufficient conditions for the invertibility of this kind of matrice...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2017 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/106362 |
| Acceso en línea: | https://hdl.handle.net/2117/106362 https://dx.doi.org/10.1016/j.laa.2017.06.010 |
| Access Level: | acceso abierto |
| Palabra clave: | Toeplitz matrices Differential equations, Linear tridiagonal matrices quasi–periodic sequences second order linear difference equations boundary value problems discrete Schrödinger operator Toeplitz, Matrius de Equacions diferencials lineals Classificació AMS::15 Linear and multilinear algebra matrix theory Classificació AMS::31 Potential theory Classificació AMS::39 Difference and functional equations::39A Difference equations Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Equacions en diferències |
| Sumario: | We have named tridiagonal (p,r)–Toeplitz matrix to those tridiagonal matrices in which each diagonal is a quasi–periodic sequence, d(p+j)=rd(j), so with period p¿N but multiplied by a real number r. We present here the necessary and sufficient conditions for the invertibility of this kind of matrices and explicitly compute their inverse. The techniques we use are related with the solution of boundary value problems associated to second order linear difference equations. These boundary value problems can be expressed throughout the discrete Schrödinger operator and their solutions can be computed using recent advances in the study of linear difference equations with quasi–periodic coefficients. The conditions that ensure the uniqueness solution of the boundary value problem lead us to the invertibility conditions for the matrix, whereas the solutions of the boundary value problems provides the entries of the inverse matrix. |
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