The determinants of certain \((0,1)\) Toeplitz matrices
type: Item
instance of: scholarly article
title: The determinants of certain \((0,1)\) Toeplitz matrices
zbMATH Open document ID: 1464.15012
published in: Linear Algebra and its Applications
publication date: 2021-04-24T00:00:00Z
review text: This paper deals with pentadiagonal Toeplitz matrices, namely square Toeplitz matrices with four subdiagonals, two above and two below the main diagonal, that are nonzero (see [\textit{M. Andelić} and \textit{C. M. da Fonseca}, ``Some determinantal considerations for pentadiagonal matrices'', Linear Multilinear Algebra (to appear), \url{doi:10.1080/03081087.2019.1708845}]). The following is the setting of this paper. Consider natural numbers \(n> t \geq 2\) and define the \(n \times n\) matrix \(A(n,t)\) by \[ [A(n,t)]_{ij} = \begin{cases} 1 & \quad\text{if } j-i \in \{-2, -1, 0, 1, t\} \\ 0 &\quad \text{otherwise.} \end{cases} \] Suppose \(\Delta(n,t) = \det A(n,t)\). The main result claims that for any fixed natural number \(k \geq 1\), the sequence \(\Delta(n, n-k)\) has period four on the range \(n \geq 2k\). In particular, this settles a question posed in the above paper.
reviewed by: Jaydeb Sarkar
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B05
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 7337633
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: determinants
zbMATH Keywords: band matrices
author: Ya. N. Shitov
MaRDI profile type: Publication
full work available at URL: j.laa.2021.02.002
OpenAlex ID: W3126695619
cites work: Explicit determinantal formula for a class of banded matrices
cites work: On a homogeneous recurrence relation for the determinants of general pentadiagonal Toeplitz matrices
cites work: Recursion formulae for the characteristic polynomial of symmetric banded matrices
cites work: On the characteristic polynomial, eigenvectors and determinant of heptadiagonal matrices
cites work: Five-Diagonal Toeplitz Determinants an Their Relation to Chebyshev Polynomials
cites work: On five-diagonal Toeplitz matrices and orthogonal polynomials on the unit circle
cites work: A recursive relation for the determinant of a pentadiagonal matrix
cites work: An explicit formula for the inverse of a pentadiagonal Toeplitz matrix
DOI: 10.1016/J.LAA.2021.02.002
description: scientific article; zbMATH DE number 7337633