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 and its last row respectively. This class of matrices encompasses both standard Jacobi and periodic Jacobi matrices that appear in many context...

Full description

Saved in:
Bibliographic Details
Published in:Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Vol. 113; no. 4; pp. 3795 - 3828
Main Authors: Encinas, A. M., Jiménez, M. J.
Format: Journal Article Publication
Language:English
Published: Cham Springer International Publishing 01.10.2019
Springer Nature B.V
Springer
Subjects:
11 Number theory
11B Sequences and sets
15 Linear and multilinear algebra; matrix theory
34 Ordinary differential equations
34B Boundary value problems
39 Difference and functional equations
39A Difference equations
Anàlisi matricial
Applications of Mathematics
Boundary value problem
Boundary value problems
Classificació AMS
Difference equations
Generalized Jacobi matrix
Matemàtiques i estadística
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Matrix methods
Original Paper
Second order difference equation
Theoretical
Àrees temàtiques de la UPC
Boundary value problem
11B83
34B45
Second order difference equation
39A06
15B05
Generalized Jacobi matrix
15A09
ISSN:1578-7303, 1579-1505
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 and its last row respectively. This class of matrices encompasses both standard Jacobi and periodic Jacobi matrices that appear in many contexts in pure and applied mathematics. Therefore, the study of the inverse of these matrices becomes of specific interest. However, explicit formulas for inverses are known only in a few cases, in particular when the coefficients of the diagonal entries are subjected to some restrictions. We will show that the inverse of generalized Jacobi matrices can be raised in terms of the resolution of a boundary value problem associated with a second order linear difference equation. In fact, recent advances in the study of linear difference equations, allow us to compute the solution of this kind of boundary value problems. So, the conditions that ensure the uniqueness of the solution of the boundary value problem leads to the invertibility conditions for the matrix, whereas that solutions for suitable problems provide explicitly the entries of the inverse matrix.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-019-00723-3