Combining triple diagonal forms
type: Item
instance of: scholarly article
title: Combining triple diagonal forms
zbMATH Open document ID: 1209.15018
author: Edward Rehkopf
published in: Linear Algebra and its Applications
publication date: 2011-01-07T00:00:00Z
review text: Let \(R\) be a principal ideal domain of characteristic not equal to 2. Two matrices \(A,C \in M_n(R)\) are said to be congruent if \(C=P^TAP\) for some \(P \in GL_n(R)\). It is known (Jacobi) that a symmetric matrix \(A \in M_n(R)\) is congruent to a tridiagonal matrix. This form was strengthened by some authors in different ways. In this paper the author works towards combining these refinements. Specifically, he proves the following main result: Let \(R\) be a Euclidean domain of characteristic not equal to 2, and let \(A\) be a symmetric matrix in \(M_n(R)\) such that \(d= \det{A} \neq 0\). Then, \(A\) is congruent to either a tridiagonal matrix \[ C=\left( \begin{matrix} c_1 & d_1 & 0 & \ldots & 0 & 0 \\ d_1 & c_2 & d_2 & \ldots & 0 & 0 \\ 0 & d_2 & c_3 & \dots & 0 & 0 \\ \vdots & \vdots &\vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \dots & c_{n-1}& d_{n-1} \\ 0 & 0 & 0 & \dots & d_{n-1}& c_n \end{matrix} \right), \] where both the following hold: {\parindent=6.5mm \begin{itemize}\item[(i)] \(|d_1| \leq |d_2| \leq \cdots \leq |d_{n-1}|\), \item[(ii)] \(d_i | d\) for \(i \leq n-4\), \end{itemize}} or a block diagonal matrix with each block satisfying this condition.
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 5833983
zbMATH Keywords: quadratic form
zbMATH Keywords: tridiagonal matrix
zbMATH Keywords: symmetric matrix
zbMATH Keywords: congruent matrices
zbMATH Keywords: lattice
zbMATH Keywords: principal ideal domain
zbMATH Keywords: Euclidean domain
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.laa.2010.09.043
OpenAlex ID: W2069397657
cites work: Unimodular quadratic forms over global function fields
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cites work: Tridiagonal matrices
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cites work: Sharpening the triple diagonal form
cites work: Q3708857
DOI: 10.1016/J.LAA.2010.09.043
description: scientific article; zbMATH DE number 5833983