Sylvester matrix equation for matrix pencils
type: Item
instance of: scholarly article
title: Sylvester matrix equation for matrix pencils
zbMATH Open document ID: 0840.15010
DOI: 10.1016/0024-3795(94)00044-1
author: Yanyan Li
published in: Linear Algebra and its Applications
publication date: 1996-07-04T00:00:00Z
review text: Let be given complex matrices \(B_1\), \(A_1 \in \mathbb{C}^{p_1 \times q_1}\) and \(B_2\), \(A_2 \in \mathbb{C}^{p_2 \times q_2}\), and consider the homogeneous system of equations \[ A_1 X - YA_2 = 0, \quad B_1 X - YB_2 = 0. \tag{1} \] The dimension of the set of solutions of (1) is obtained in terms of the Kronecker invariants of the matrix pencils \(\lambda B_1 - A_1\) and \(\lambda B_2 - A_2\). Necessary and sufficients conditions for strict equivalence of matrix pencils are given in terms of ranks of several matrices, in particular, equality of ranks of certain \(2 \times 2\) block matrices, where each block is a Kronecker product. These conditions represent a generalization of the matrix similarity criterion given by \textit{S. Friedland} [Lect. Appl. Math. 18, AMS, 43-85 (1980; MR 82b:15014)]. Analogous necessary and sufficient conditions are given for existence of solutions of the inhomogeneous system \(A_1 X - YA_2 = A_3\), \(B_1X - YB_2 = B_3\).
reviewed by: Leiba Rodman
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A22
Mathematics Subject Classification ID: 15A06
zbMATH DE Number: 841760
zbMATH Keywords: intertwining linear equations
zbMATH Keywords: Kronecker invariants
zbMATH Keywords: matrix pencils
zbMATH Keywords: ranks
zbMATH Keywords: Kronecker product
zbMATH Keywords: matrix similarity
zbMATH Keywords: MR 82b:15014
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00044-1
OpenAlex ID: W1977410622
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description: scientific article; zbMATH DE number 841760