Absolutely indecomposable symmetric matrices
type: Item
instance of: scholarly article
title: Absolutely indecomposable symmetric matrices
zbMATH Open document ID: 1012.15010
DOI: 10.1016/S0022-4049(02)00037-3
author: H. Ochsenius
author: H. A. Keller
published in: Journal of Pure and Applied Algebra
publication date: 2002-12-02T00:00:00Z
review text: It is well known that every real symmetric matrix can be diagonalized by means of conjugation with an orthogonal matrix. But this is not the case for matrices over arbitrary fields. The authors focus on the more general problem, i.e., the decomposability of a given symmetric matrix into smaller blocks orthogonally. A symmetric matrix \(A\) is called absolutely indecomposable if it is indecomposable over every extension of the base field. This can only happen if the base field \(K\) is a non-formally real field. Assume that \(K\) is such a field of level \(s\) (i.e. \(s\) is the smallest integer such that \(-1\) is a sum of \(s\) squares in \(K\)), then it is proved that in Mat\(_n(K)\) there exist symmetric absolutely indecomposable matrices iff \(n>s\) and \(n\equiv 0\pmod {2s}\) or \(n\equiv \pm 1\pmod {2s}\). The proofs yield an explicit description of such matrices. On the other hand, if \(n\leq s\), it is shown that every symmetric matrix can be diagonalized orthogonally over some extension field of \(K\).
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 1839049
zbMATH Keywords: symmetric matrix
zbMATH Keywords: indecomposable matrix
zbMATH Keywords: diagonalization
MaRDI profile type: Publication
cites work: Q3356452
cites work: Q4771385
cites work: Q3882586
full work available at URL: s0022-4049(02)00037-3
OpenAlex ID: W1976080430
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description: scientific article; zbMATH DE number 1839049