Explicit Inverse Of Symmetric, Tridiagonal Near Toeplitz Matrices With Strictly Diagonally Dominant Toeplitz Part
Document Type
Article
Source of Publication
Special Matrices
Publication Date
1-1-2025
Abstract
Let T n = tridiag ( - 1 , b , - 1 ) , an n x n symmetric, strictly diagonally dominant tridiagonal matrix ( divided by b divided by > 2 ). This article investigates tridiagonal near-Toeplitz matrices T n & colone; [ t i , j ] , obtained by perturbing the ( 1 , 1 ) and ( n , n ) entry of T n . Let t 1 , 1 = t n , n = b not equal b . We derive exact inverses of T n . Furthermore, we demonstrate that these results hold even when divided by b divided by < 1 . Additionally, we establish upper bounds for the infinite norms of the inverse matrices. The row sums and traces of the inverse provide insight into the matrix's spectral properties and play a key role in understanding the convergence of fixed-point iterations. These metrics allow us to derive tighter bounds on the infinite norms and improve computational efficiency. Numerical results for Fisher's problem demonstrate that the derived bounds closely match the actual infinite norms, particularly for b > 2 with b <= 1 and b < - 2 with b >= - 1 . For other cases, further refinement of the bounds is possible. Our results contribute to improving the convergence rates of fixed-point iterations and reducing the computation time for matrix inversion.
DOI Link
ISSN
Publisher
Walter de Gruyter GmbH
Volume
13
Issue
1
Disciplines
Mathematics
Keywords
Toeplitz matrices, diagonal dominance, exact inverses, upper bounds
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Kurmanbek, Bakytzhan; Erlangga, Yogi; and Amanbek, Yerlan, "Explicit Inverse Of Symmetric, Tridiagonal Near Toeplitz Matrices With Strictly Diagonally Dominant Toeplitz Part" (2025). All Works. 7118.
https://zuscholars.zu.ac.ae/works/7118
Indexed in Scopus
no
Open Access
yes
Open Access Type
Gold: This publication is openly available in an open access journal/series
