Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices
We have named generalized Jacobi matrices to those that are practically tridiagonal, except for the two final entries and the two first entries of its first andits last row respectively. This class of matrices encompasses both standard Jacobiand periodic Jacobi matrices that appear in many contexts...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2019 |
| 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:2117/168258 |
| Acceso en línea: | https://hdl.handle.net/2117/168258 https://dx.doi.org/10.1007/s13398-019-00723-3 |
| Access Level: | acceso abierto |
| Palabra clave: | Second order difference equation Boundary value problem Generalized Jacobi matrix Anàlisi matricial Classificació AMS::34 Ordinary differential equations::34B Boundary value problems Classificació AMS::39 Difference and functional equations::39A Difference equations Classificació AMS::15 Linear and multilinear algebra matrix theory Classificació AMS::11 Number theory::11B Sequences and sets Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | We have named generalized Jacobi matrices to those that are practically tridiagonal, except for the two final entries and the two first entries of its first andits last row respectively. This class of matrices encompasses both standard Jacobiand periodic Jacobi matrices that appear in many contexts in pure and appliedmathematics. Therefore, the study of the inverse of these matrices becomes ofspecific interest. However, explicit formulas for inverses are known only in a fewcases, in particular when the coefficients of the diagonal entries are subjected tosome restrictions.We will show that the inverse of generalized Jacobi matrices can be raisedin terms of the resolution of a boundary value problem associated with a secondorder linear difference equation. In fact, recent advances in the study of lineardifference equations, allow us to compute the solution of this kind of boundaryvalue problems. So, the conditions that ensure the uniqueness of the solution ofthe boundary value problem leads to the invertibility conditions for the matrix,whereas that solutions for suitable problems provide explicitly the entries of theinverse matrix. |
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