Cyclizable matrix pairs over \(\mathbb C[x]\) and a conjecture on Toeplitz pencils
type: Item
instance of: scholarly article
title: Cyclizable matrix pairs over \(\mathbb C[x]\) and a conjecture on Toeplitz pencils
zbMATH Open document ID: 1068.15008
author: Wiland Schmale
author: Pramod K. Sharma
published in: Linear Algebra and its Applications
publication date: 2004-11-22T00:00:00Z
review text: Given nonnegative integers \(n,m\) and a commutative ring \(R\), a matrix pair \((A,B)\) with \(A\in R^{n\times n}\), \(B\in R^{n\times m}\) is called reachable if \([B,AB,\dots,A^{n-1}B]\) is right invertible. The cyclization property claims that any reachable matrix pair is cyclizable, i.e., there exist \(u\in R^ m\), \(F\in R^{n\times m}\) such that \([Bu, (A+BF)Bu,\dots, (A+BF)^{n-1}Bu]\) is invertible. It is shown that for a whole class of pairs over \({\mathbb C}[x]\) the cyclization property holds if a certain conjecture on Toeplitz pencils of matrices over \({\mathbb C}[x]\) is fulfilled. An algebraic computable test for the validity of the latter is given. Based on these results the validity of the conjecture on cyclizability is show to be true up to dimension \(n=5\).
reviewed by: Hans Havlicek
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A22
zbMATH DE Number: 2117280
zbMATH Keywords: systems over rings
zbMATH Keywords: cyclizability
zbMATH Keywords: Toeplitz pencils
zbMATH Keywords: BSSV-conjecture
zbMATH Keywords: reachable matrix pair
describes a project that uses: SINGULAR
MaRDI profile type: Publication
Wikidata QID: Q122980785
full work available at URL: j.laa.2004.03.033
OpenAlex ID: W2028921559
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DOI: 10.1016/J.LAA.2004.03.033
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description: scientific article; zbMATH DE number 2117280