Normal forms of generic triangular band matrices and Jordan forms of nilpotent completions
type: Item
instance of: scholarly article
title: Normal forms of generic triangular band matrices and Jordan forms of nilpotent completions
zbMATH Open document ID: 0963.15009
DOI: 10.1016/S0024-3795(99)00278-5
author: Leiba Rodman
author: Michael Gekhtman
published in: Linear Algebra and its Applications
publication date: 2001-06-18T00:00:00Z
review text: Let \(F\) be an infinite field. For fixed integers \(k\) and \(n\), where \(1<k\leq n\), consider the set of triangular band matrices \({\mathcal D}_{k,n}\) formed by all the \(n\times n\) matrices \(A=[a_{ij}]_{i,j=1}^n\in F^{n\times n}\) satisfying the conditions: (1) \(a_{ij}=0\) for \(i>j\), (2) \(a_{ij}=0\) for \(j-i\geq k\), (3) \(a_{i,i+k-1}\not= 0\) for \(i=1,2,\dots ,n-k+1\). The authors study the possible Jordan forms of the nilpotent \(n\times n\) matrices with a given upper triangular part \(A\in {\mathcal D}_{k,n}\) with zero trace, that is, the nilpotent matrices of the form \(A+T\), where \(T=[t_{ij}]_{i,j=1}^n\in F^{n\times n}\) is strictly lower triangular (\(t_{ij}=0\) for \(i\leq j\)). For generic matrices within \({\mathcal D}_{k,n}\), the set of Jordan forms contains a unique minimal Jordan form. Moreover, in the generic case, there exists a nilpotent matrix in the orbit of a given upper triangular matrix having the minimal Jordan form, and also having the minimal rank among all matrices with the given upper triangular part. The orbit is understood in the sense of the triangular group action.
reviewed by: Nicolae Cotfas
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 1443395
zbMATH Keywords: triangular matrices
zbMATH Keywords: triangular group action
zbMATH Keywords: nilpotent completions
zbMATH Keywords: normal forms
zbMATH Keywords: Jordan forms
zbMATH Keywords: minimal rank
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1443395