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

Zhaolin Jiang,Seongtae Jhang,Yaru Fu,Xiaoyu Jiang

doi:10.1515/spma-2020-0012

Zhaolin Jiang, Seongtae Jhang + Show 2 more

Open Access

https://doi.org/10.1515/spma-2020-0012

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Journal: Special Matrices	Publication Date: May 4, 2020
Citations: 7	License type: CC BY 4.0

Affiliation: Linyi University, University of Suwon

Abstract

AbstractIn 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.

Highlights

We start by introducing the main research object of this paper
We present some algorithms based on the main theoretical results
We show the superiority of our method by comparing the CPU time of some existing algorithms studied recently

Summary

Introduction

An n × n Toeplitz tridiagonal matrix with perturbed columns A(c , an; d, −d, −d; a , cn), shorted for A, with Ai, = ai for i ∈ { , n}, Ai,n = ci for i ∈ { , n} and Ai,j = −d for j − ≤ i ≤ j (i ≠ n), ≤ j ≤ n, Ai,i− = d for ≤ i ≤ n and 0 otherwise. We can obtain much more e cient and simple theoretical results and algorithms for computing the determinants and inverses of Toeplitz tridiagonal matrices with perturbed columns. Using the sparsity of the Toeplitz and Hankel tridiagonal matrices with perturbed columns, the famous Sherman-Morrison-Woodbury formula, the explicit expression of the inverse matrix can be obtained.

Results

Conclusion