Simultaneous reduction to triangular forms after extension with zeroes
type: Item
instance of: scholarly article
title: Simultaneous reduction to triangular forms after extension with zeroes
zbMATH Open document ID: 0941.15004
DOI: 10.1016/S0024-3795(98)10036-8
author: R. A. Zuidwijk
author: Harm Bart
published in: Linear Algebra and its Applications
publication date: 2000-07-31T00:00:00Z
review text: Setting out from the known Schur's lemma and \textit{N. H. McCoy}'s theorem [Bull. Am. Math. Soc. 42, 592-600 (1936; Zbl 0015.05501)] that provides a necessary and sufficient condition so that two complex matrices \(A\), \(Z\) admit simultaneous reduction to upper triangular form (that is, if there exists an invertible matrix \(S\), such that both \(S^{-1}AS\) and \(S^{-1}ZS\) are upper triangular), the paper studies when two complex matrices \(A\), \(Z\) admit simultaneous reduction to complementary triangular forms (that is, if there exists an invertible matrix \(S\), such that \(S^{-1}AS\) is upper triangular and \(S^{-1}ZS\) is lower triangular). This notion is originated from mathematical systems theory. The authors provide necessary and sufficient conditions for simultaneous reduction to complementary triangular forms in the special case of almost diagonalizable matrices, this is, matrices whose spectral polynomial is of rank of \(1\). Also, simultaneous reduction to complementary triangular forms after extension with zeros is discussed for pairs of nilpotent and Jordan matrices.
reviewed by: M. Ladra
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 1370028
zbMATH Keywords: maximal nest subspaces
zbMATH Keywords: simultaneous triangularization
zbMATH Keywords: extension with zeros
zbMATH Keywords: nilpotent matrices
zbMATH Keywords: almost diagonalizable matrix
zbMATH Keywords: simultaneous reduction to upper triangular form
zbMATH Keywords: spectral polynomial
zbMATH Keywords: simultaneous reduction to complementary triangular forms
zbMATH Keywords: Jordan matrices
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1370028