A generalization of Sourour's theorem
type: Item
instance of: scholarly article
title: A generalization of Sourour's theorem
zbMATH Open document ID: 0937.15007
DOI: 10.1016/S0024-3795(98)10167-2
author: N. L. Gordeev
author: Erich W. Ellers
published in: Linear Algebra and its Applications
publication date: 2000-06-13T00:00:00Z
review text: \textit{Sourour} [Linear Multilinear Algebra 19, 141-147 (1986; Zbl 0591.15008)] proved the following theorem: Let \(A\) be a nonscalar invertible matrix over a field \(K\), then there exist matrices \(B\), \(C\) with prescribed eigenvalues such that \(A=BC\). This result can be reformulated in the following two ways: (A) For every noncentral invertible matrix there is a matrix in its \(SL_n\)-conjugacy class which has a Gauss decomposition with prescribed diagonal part, and (B) For every noncentral matrix \(X\in GL_n (K)\), \(n>1\), and every sequence \((a_1, \dots, a_{n-1})\), \(a_i\in K\setminus 0\), there is a matrix \(Y\in SL_n(K)\) such that the \(i\)-th principal minor of \(YXY^{-1}\) has the value \(a_i\), \(1\leq i<n\). (A) was extended to Chevalley groups by the authors, and the purpose of this note is to remove the restriction ``\(a_i\neq 0\) for all \(i\)'' in (B).
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 15A29
zbMATH DE Number: 1331732
zbMATH Keywords: inverse problems
zbMATH Keywords: invertible matrix
zbMATH Keywords: prescribed eigenvalues
zbMATH Keywords: Gauss decomposition
zbMATH Keywords: prescribed diagonal part
zbMATH Keywords: Chevalley groups
reviewed by: Rabe von Randow
MaRDI profile type: Publication
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cites work: Gauss decomposition with prescribed semisimple part in chevalley groups iii: Finite twisted groups
cites work: On the conjectures of J. Thompson and O. Ore
cites work: Covering numbers for Chevalley groups
cites work: Products of cyclic conjugacy classes in the groups \(\text{PSL}(n,F)\)
cites work: The covering number of the group \(\text{PSL}_ n(F)\)
cites work: A factorization theorem for matrices
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description: scientific article; zbMATH DE number 1331732