Tridiagonal matrices
type: Item
instance of: scholarly article
title: Tridiagonal matrices
zbMATH Open document ID: 0811.15013
DOI: 10.1016/0024-3795(94)90103-1
author: Morris Newman
published in: Linear Algebra and its Applications
publication date: 1994-08-15T00:00:00Z
review text: Let \(A\) be an \(n \times n\) symmetric matrix with elements in a principal ideal domain \(R\). The author proves that if \(\text{det} A = d \neq 0\) then \(A\) is congruent to a tridiagonal matrix \[ \left[ \begin{matrix} c_ 1 & d_ 1 & & & \;\\ d_ 1 & c_ 2 & d_ 2 & & \;\\ & \ddots & \ddots & \ddots & \;\\ & & d_{n-2} & c_{n-1} & d_{n-1} \\ & & & d_{n-1} & c_ n \end{matrix} \right] \] where \(d_ i | d\), \(1 \leq i \leq n-2\). He also proves that if \(A\) is unimodular, then the off diagonal elements may be chosen so that \(d_ i=1\), \(1 \leq i \leq n - 2\).
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 614169
zbMATH Keywords: congruence
zbMATH Keywords: symmetric matrix
zbMATH Keywords: principal ideal domain
zbMATH Keywords: tridiagonal matrix
reviewed by: Fergus J. Gaines
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90103-1
OpenAlex ID: W4206271192
cites work: Positive Definite Matrices and Catalan Numbers
cites work: Q5668937
description: scientific article; zbMATH DE number 614169