ON INVERSES AND EIGENPAIRS OF PERIODIC TRIDIAGONAL TOEPLITZ MATRICES WITH PERTURBED CORNERS
Yunlan Wei, Xiaoyu Jiang, Zhaolin Jiang, Sugoog Shon
Abstract
In this paper, we derive explicit determinants, inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners of Type $I$. The Mersenne numbers play an important role in these explicit formulas derived. Our main approaches include clever uses of the Schur complement and matrix decomposition with the Sherman-Morrison-Woodbury formula. Besides, the properties of Type $II$ matrix can be also obtained, which benefits from the relation between Type $I$ and $II$ matrices. Lastly, we give three algorithms for these basic quantities and analyze them to illustrate our theoretical results.
Topics & Concepts
Toeplitz matrixTridiagonal matrixMathematicsSchur complementMersenne primePure mathematicsMatrix (chemical analysis)Algebra over a fieldType (biology)Applied mathematicsCombinatoricsEigenvalues and eigenvectorsComposite materialBiologyEcologyQuantum mechanicsMaterials sciencePhysicsMatrix Theory and AlgorithmsAdvanced Mathematical Theories and ApplicationsLiquid Crystal Research Advancements