On Fiedler's characterization of tridiagonal matrices over arbitrary fields
type: Item
instance of: scholarly article
title: On Fiedler's characterization of tridiagonal matrices over arbitrary fields
zbMATH Open document ID: 1074.15030
author: Américo Bento
author: António Leal Duarte
published in: Linear Algebra and its Applications
publication date: 2005-06-01T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/11425
review text: An application of the general theorem, treated as Fiedler's characterization of tridiagonal matrices [\textit{M. Fiedler}, Linear Algebra Appl. 2, 191--197 (1969; Zbl 0194.06105)], to the case of tridiagonal \(5\times 5\)-matrices is considered. This theorem (Fiedler's property) is formulated in the paper as follows: ``Let \(A\) be an \(n\)-by-\(n\) real diagonal matrix. Then \(\text{rank}(A+ D)\geq n- 1\), for every \(n\)-by-\(n\) real diagonal matrix \(D\), if and only if \(A\) is permutational similar to an irreducible tridiagonal matrix''. As a result, some peculiarities, extensions and limitations, related to the basic relation \(A\to P^TTP\) in the case under consideration, where \(P\) is a permutation matrix and \(T\) an irreducible tridiagonal matrix, are presented. They are obtained and argued by making direct use of the explicit expressions for a wide set of the above mentioned \(5\times 5\)-matrices and the corresponding submatrices.
reviewed by: A. A. Bogush
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 2173165
zbMATH Keywords: tridiagonal \(5\times 5\)-matrices
zbMATH Keywords: rank
zbMATH Keywords: Fiedler property
zbMATH Keywords: similarity
zbMATH Keywords: finite fields
zbMATH Keywords: completion problem
MaRDI profile type: Publication
OpenAlex ID: W1995605974
cites work: A characterization of tridiagonal matrices
cites work: Q3487569
cites work: Graph theoretic methods for matrix completion problems
cites work: On a characterization of tridiagonal matrices by M. Fiedler
DOI: 10.1016/J.LAA.2005.01.026
Recommended article: Quasi-triangularization of matrix polynomials over arbitrary fields
Recommended article: On diagonalization of matrices in an arbitrary field
Recommended article: Q3767781
Recommended article: Some asymptotics for triangular matrices over finite fields
Recommended article: On characterization of tripotent matrices in triangular matrix rings
Recommended article: Q3610092
Recommended article: On some Toeplitz-type matrices over finite fields
Recommended article: Q4284186
Recommended article: Q3741737
Recommended article: Traces of powers of matrices over finite fields
description: scientific article; zbMATH DE number 2173165