Miniversal deformations of marked matrices
type: Item
instance of: scholarly article
title: Miniversal deformations of marked matrices
zbMATH Open document ID: 1038.15008
DOI: 10.1016/S0024-3795(02)00390-7
author: Albert Compta
author: Josep Ferrer
author: Ferran Puerta
published in: Linear Algebra and its Applications
publication date: 2003-03-11T00:00:00Z
review text: For fixed positive integers \(m\) and \(n\), let \(\mathcal{M}\) be the set of all block matrices over \(\mathbb{C}\) of the form \(a:=\left[ \begin{smallmatrix} A & C\\ 0 & B \end{smallmatrix} \right] \) where \(A\) is \(n\times n\), \(C\) is \(n\times m\) and \(B\) is \(m\times m\), and let \(\mathcal{G}\) be the group of invertible matrices of this form. The authors show that a miniversal deformation of this matrix is determined by a linear submanifold \(a+N\) where \(N\) consists of all matrices \(x:=\left[ \begin{smallmatrix} X & Z\\ 0 & Y \end{smallmatrix} \right] \) in \(\mathcal{M}\) satisfying: (1) \(A^{\ast}Z=ZB^{\ast}\); (2) \(\left[ A^{\ast},X\right] =ZC^{\ast}\); and (3) \(\left[ Y,B^{\ast}\right] =C^{\ast}Z\). They then characterize \(N\) more precisely in the case that \(a\) is a marked nilpotent matrix (that is, \(a\) is similar under \(\mathcal{G}\) to a nilpotent Jordan matrix). The final section shows how these results can be applied to the Carlson problem [see \textit{A. Compta} and \textit{J. Ferrer}, Matricial realizations of the solutions of the Carlson problem, Linear Algebra Appl. 353, No. 1--3, 197-206 (2002; Zbl 1055.15015)].
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A30
zbMATH DE Number: 1880055
zbMATH Keywords: marked matrices
zbMATH Keywords: block matrices
zbMATH Keywords: miniversal deformation
zbMATH Keywords: nilpotent matrix
zbMATH Keywords: Carlson problem
MaRDI profile type: Publication
cites work: ON MATRICES DEPENDING ON PARAMETERS
cites work: Extensions of Jordan bases for invariant subspaces of a matrix
cites work: A geometric approach to the Carlson problem
cites work: Matricial realizations of the solutions of the Carlson problem
cites work: Brunowsky local form of a holomorphic family of pairs of matrices
cites work: Differentiable families of subspaces
cites work: Versal deformations of invariant subspaces
cites work: Eigenvalues, invariant factors, highest weights, and Schubert calculus
cites work: Q3748420
cites work: Q4744461
cites work: The Multiplication of Schur-Functions and Extensions of p -Modules
cites work: Stable bundles, representation theory and Hermitian operators
cites work: The honeycomb model of $GL_n(\mathbb C)$ tensor products I: Proof of the saturation conjecture
cites work: Invariance and system theory: algebraic and geometric aspects
cites work: Q5651936
cites work: Topological classification and structural stability of linear systems
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description: scientific article; zbMATH DE number 1880055