Nilpotent completions of partial upper triangular matrices
type: Item
instance of: scholarly article
title: Nilpotent completions of partial upper triangular matrices
zbMATH Open document ID: 1061.15015
published in: Linear Algebra and its Applications
publication date: 2004-11-18T00:00:00Z
review text: The paper deals with different completion problems with partial upper triangular matrices. Specifically, the author presents and solves the following questions: (a) Given a partial upper triangular matrix \(A\), lower irreducible and with trace equal to zero, does there exist a nilpotent completion \(A_c\) of \(A\), such that its rank is equal to the minimal rank of \(A\)? (b) Let \(r_N\) be the minimun of the rank of all nilpotent completions of a partial matrix \(A\). If \(A\) is a partial upper triangular matrix, lower irreducible and with trace equal to zero, what is the value of \(r_N\)? (c) Let \(A\) be a partial upper triangular matrix, lower irreducible and with trace equal to zero. If \(r(A)\) denote the minimal rank of \(A\), are the following inequalities satisfied: \[ r(A^k)-r(A^{k+1}) \geq r(A^{k+1})- r(A^{k+2}), \forall k \geq 1 ? \] (d) Let \(A\) be a partial upper triangular matrix, lower irreducible and with trace equal to zero. Then, does there exist a nilpotent completion of \(A\) such that its Segre characteristic is majorized by the Segre characteristic of all possible nilpotent completion of \(A\)?
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 2116534
zbMATH Keywords: completion problem
zbMATH Keywords: partial matrix
zbMATH Keywords: minimal rank
zbMATH Keywords: Jordan form
zbMATH Keywords: partial upper triangular matrices
zbMATH Keywords: nilpotent completions
zbMATH Keywords: irreducible
zbMATH Keywords: trace
zbMATH Keywords: inequalities
zbMATH Keywords: Segre characteristic
author: Ana M. Urbano
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.laa.2004.04.018
OpenAlex ID: W2073897468
cites work: Q3831127
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cites work: On the Jordan form of completions of partial upper triangular matrices
cites work: An algorithm for nilpotent completions of partial Jordan matrices
cites work: Q3838054
cites work: Jordan structures of strictly lower triangular completions of nilpotent matrices
cites work: Jordan forms of completions of partial upper triangular matrices
cites work: Minimal rank completions for block matrices
DOI: 10.1016/J.LAA.2004.04.018
description: scientific article; zbMATH DE number 2116534