Matrix pencils completion problems. II
type: Item
instance of: scholarly article
title: Matrix pencils completion problems. II
zbMATH Open document ID: 1148.15008
author: Marija Dodig
published in: Linear Algebra and its Applications
publication date: 2008-06-17T00:00:00Z
review text: [For part I see the author, ibid. 428, No.~1, 259--304 (2008; Zbl 1131.15010).] The paper deals with the matrix pencils completion problem. This problem analyzes the existence of a matrix pencil with prescribed Kronecker invariants and a regular subpencil. Specifically, the author studies the following problem: Let \(A(\lambda) \in F[\lambda]^{n \times n}\) be a regular matrix pencil. Let \(E(\lambda) \in F[\lambda]^{(n+x) \times (n+y)}\) be a matrix pencil. Find necessary and sufficient conditions for the existence of pencils \(B(\lambda) \in F[\lambda]^{n \times x}\), \(C(\lambda) \in F[\lambda]^{y \times n}\) and \(D(\lambda) \in F[\lambda]^{y \times x}\) such that the pencil \[ \left( \begin{matrix} A(\lambda) & B(\lambda) \\ C(\lambda) & D(\lambda) \end{matrix} \right) \] is strictly equivalent to \(E(\lambda)\). The author explicitly solves the above problem, by giving necessary and sufficient conditions without any existential quantifiers involved. On the other hand, in the second part of this paper the following problem is analyzed: Let \(A(\lambda) \in F[\lambda]^{(n+p) \times (n+m)}\) and \(E(\lambda) \in F[\lambda]^{(n+p+m+x) \times (n+m+p+y)}\) be matrix pencils. Find necessary and sufficient conditions for the existence of pencils \(B(\lambda) \in F[\lambda]^{(n+p) \times (p+x)}\), \(C(\lambda) \in F[\lambda]^{(m+y) \times (n+m)}\) and \(D(\lambda) \in F[\lambda]^{(m+y) \times (x+p)}\) such that the pencil \[ M(\lambda)=\left( \begin{matrix} A(\lambda) & B(\lambda) \\ C(\lambda) & D(\lambda) \end{matrix}\right) \] is strictly equivalent to \(E(\lambda)\) and such that \(\text{rank}[A(\lambda) \;\;B(\lambda)]=n+p\) and \(\text{rank}[A(\lambda) \;\;C(\lambda)]^T=n+m\). The author gives a complete solution of the above problem and she proves that this problem is equivalent to the problem of describing the possible Kronecker invariants of a pencil \(M(\lambda)\) with a prescribed arbitrary subpencil \(A(\lambda)\), if there exists a regular intermediate pen…
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A29
zbMATH DE Number: 5289140
zbMATH Keywords: matrix pencils
zbMATH Keywords: regular subpencil
zbMATH Keywords: completion problem
zbMATH Keywords: prescribed Kronecker invariants
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.laa.2008.03.021
OpenAlex ID: W2047263968
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DOI: 10.1016/J.LAA.2008.03.021
description: scientific article; zbMATH DE number 5289140