Miniversal deformations of marked matrices
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)].
scientific article; zbMATH DE number 1880055
scientific article