Some comments on k -tridiagonal matrices: Determinant, spectra, and inversion

“…Formulas (26) and (27) would give an analytic formula for A −1 . However, there is a big advantage of (12) from computational consideration as we shall see from Section 3.…”

Section: Remarkmentioning

confidence: 99%

“…On the other hand, many studies have been conducted for tridiagonal matrices or periodic tridiagonal matrices, especially for their determinants and inverses [22][23][24][25][26][27][28][29][30]. Two decades ago, Wittenburg [31] studied the inverse of tridiagonal toeplitz and periodic matrices and applied them to elastostatics and vibration theory.…”

Section: Introductionmentioning

confidence: 99%

A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers

et al. 2019

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In this paper, we study periodic tridiagonal Toeplitz matrices with perturbed corners. By using some matrix transformations, the Schur complement and matrix decompositions techniques, as well as the Sherman-Morrison-Woodbury formula, we derive explicit determinants and inverses of these matrices. One feature of these formulas is the connection with the famous Mersenne numbers. We also propose two algorithms to illustrate our formulas.

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“…In addition, Eigenpairs, determinants and inverses of the tridiagonal matrices, anti-tridiagonal Hankel matrices, pentadiagonal matrices and cyclic pentadiagonal matrices with Toeplitz structure have been studied in [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The determinant and inversion of A(c , an; d k , a k , c k ; a , cn) have been studied extensively and found with simple and analytic expression (see [24][25][26]).…”

Section: Introductionmentioning

confidence: 99%

Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns

Jiang

et al. 2020

Special Matrices

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In this paper, our main attention is paid to calculate the determinants and inverses of two types Toeplitz and Hankel tridiagonal matrices with perturbed columns. Specifically, the determinants of the n × n Toeplitz tridiagonal matrices with perturbed columns (type I, II) can be expressed by using the famous Fibonacci numbers, the inverses of Toeplitz tridiagonal matrices with perturbed columns can also be expressed by using the well-known Lucas numbers and four entries in matrix 𝔸. And the determinants of the n×n Hankel tridiagonal matrices with perturbed columns (type I, II) are (−1]) {\left( { - 1} \right)^{{{n\left( {n - 1} \right)} \over 2}}} times of the determinant of the Toeplitz tridiagonal matrix with perturbed columns type I, the entries of the inverses of the Hankel tridiagonal matrices with perturbed columns (type I, II) are the same as that of the inverse of Toeplitz tridiagonal matrix with perturbed columns type I, except the position. In addition, we present some algorithms based on the main theoretical results. Comparison of our new algorithms and some recent works is given. The numerical result confirms our new theoretical results. And we show the superiority of our method by comparing the CPU time of some existing algorithms studied recently.

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Some comments on k -tridiagonal matrices: Determinant, spectra, and inversion

Cited by 28 publications

References 17 publications

Eigenpairs of some particular band Toeplitz matrices: A comment

Eigenpairs of some particular band Toeplitz matrices: A comment

A Study of Determinants and Inverses for Periodic Tridiagonal Toeplitz Matrices with Perturbed Corners Involving Mersenne Numbers

Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns

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