Matrices with prescribed rows and invariant factors
type: Item
instance of: scholarly article
title: Matrices with prescribed rows and invariant factors
zbMATH Open document ID: 0632.15003
DOI: 10.1016/0024-3795(87)90162-5
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://dialnet.unirioja.es/servlet/oaiart
review text: This paper contains a most interesting and beautiful result. About nine years ago \textit{E. M. de Sá} [ibid. 24, 33-50 (1979; Zbl 0395.15009)] and \textit{R. C. Thompson} [ibid. 24, 1-31 (1979; Zbl 0395.15003)] found the complete relations between the invariant polynomials of a matrix (over a field) and the invariant polynomials of one of its principal submatrices. Their result is known as Sá-Thompson interlacing inequalities and raised some questions so far unsettled. The present author tackles and solves another problem of the same type: consider the matrix (over a field) \[ G=\left[ \begin{matrix} A\quad B\\ C\quad D\end{matrix} \right] \] where A and D are square blocks, A and B are known and C and D are unknown. Find a necessary and sufficient condition for the existence of C and D so that G has prescribed invariant polynomials. It cannot be said that the condition presented in this paper is too involved to be stated here. What prevents the reviewer from stating it is the need of explaining the notation. However the reviewer would like to mention that the necessary and sufficient condition consists of some interlacing inequalities similar to those of Sá and Thompson (as expected) together with a majorization relation involving the controllability indices of the pair (A,B) and the degrees of certain polynomials.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 4027674
zbMATH Keywords: prescribed rows
zbMATH Keywords: invariant factors
zbMATH Keywords: invariant polynomials
zbMATH Keywords: Sá-Thompson interlacing inequalities
zbMATH Keywords: controllability indices
author: Ion Zaballa
MaRDI profile type: Publication
OpenAlex ID: W2081023948
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description: scientific article; zbMATH DE number 4027674