An extension of a factorization theorem of Wedderburn to matrix rings
type: Item
instance of: scholarly article
title: An extension of a factorization theorem of Wedderburn to matrix rings
zbMATH Open document ID: 0760.15009
DOI: 10.1016/0024-3795(92)90029-A
author: Thomas J. Laffey
author: Eleanor Meehan
published in: Linear Algebra and its Applications
publication date: 1992-09-27T00:00:00Z
review text: Let \(R\) be a ring with identity, \(Z(R)\) the center of \(R\) and \(f(x)\in Z(R)[x]\) a monic polynomial of degree \(n\). Suppose \(f(a) = 0\) for some \(a \in R\). The authors show that under certain assumptions \(f(x) = (x-a_ 1)\cdots (x-a_ n)\). They develop an inductive formula to obtain the \(a_ i\) in \(R\). The procedure also works in a matrix situation and yields the following theorem. Let \(F\) be a field of characteristic zero and \(f(x) \in F[x]\) a monic polynomial of degree \(n\). Then there are nonderogatory \(n\times n\) matrices \(A_ 1,\dots, A_ n\) with entries in \(F\), all having characteristic polynomial \(f(x)\) such that \(f(x)I_ n = (xI_ n - A_ 1)\cdots (xI_ n - A_ n)\).
reviewed by: Erich W. Ellers
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A23
zbMATH DE Number: 64491
zbMATH Keywords: factorization
zbMATH Keywords: matrix rings
zbMATH Keywords: characteristic polynomial
MaRDI profile type: Publication
cites work: Q5569204
cites work: Factorization of polynomials using commuting matrices
cites work: Factorization of polynomials using nilpotent Jordan blocks
cites work: Q4106341
cites work: Q3910639
full work available at URL: 0024-3795(92)90029-a
OpenAlex ID: W2054954148
Wikidata QID: Q127533163
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description: scientific article; zbMATH DE number 64491