Upper triangular similarity of upper triangular matrices
type: Item
instance of: scholarly article
title: Upper triangular similarity of upper triangular matrices
zbMATH Open document ID: 0881.15011
DOI: 10.1016/S0024-3795(97)80007-9
author: Philip Thijsse
published in: Linear Algebra and its Applications
publication date: 1997-07-28T00:00:00Z
review text: The author considers the following equivalence relation in the set of all complex upper triangular \(n\times n\) matrices: \(A\) and \(B\) are \(U\)-similar if there exists an invertible upper triangular matrix \(S\) such that \(A=S^{-1} BS\). If \(A\) and \(B\) are \(U\)-similar, then they must have the same diagonal and the same Jordan form. It is known that for \(n\geq 6\) there are infinitely many pairwise non-\(U\)-similar nilpotent upper triangular matrices with the same Jordan form. The author generalizes the concept of Jordan block (called irreducible) and proves that an upper triangular matrix is \(U\)-similar to a ``generalized'' direct sum of irreducible blocks, where the location and order of the blocks is fixed and each block is uniquely determined up to \(U\)-similarity.
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 1040491
zbMATH Keywords: upper triangular similarity
zbMATH Keywords: upper triangular matrices
zbMATH Keywords: Jordan form
reviewed by: Rabe von Randow
MaRDI profile type: Publication
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full work available at URL: s0024-3795(97)80007-9
OpenAlex ID: W2047155776
description: scientific article; zbMATH DE number 1040491