On the matrix equation \(AX=B\) with applications to the generators of a controllability matrix
type: Item
instance of: scholarly article
title: On the matrix equation \(AX=B\) with applications to the generators of a controllability matrix
zbMATH Open document ID: 0612.15006
DOI: 10.1016/0024-3795(87)90217-5
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
review text: The following results are obtained concerning the rank of the solutions of the matrix equation \(AX=B\) where A is \(m\times n\) and B is \(m\times p\). 1. There exist solutions of all ranks q with rank \(B\leq q\leq \min (n+rank B-rank A,p)\). 2. If there exists a solution \(X^*\) with rank \(X^*=\ell\) then there exist solutions of all ranks q with \(\ell -r\leq q\leq \min (p,\ell +k-r)\) where \(k=\dim \ker A\) and \(r=\dim (\ker A\cap range X)\). These results are applied to determine when a given \(n\times nm\) matrix \({\mathcal A}\) occurs as a controllability matrix of a pair (A,B).
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 93B05
zbMATH DE Number: 3989561
zbMATH Keywords: rank
zbMATH Keywords: matrix equation
zbMATH Keywords: controllability matrix
author: Frank Uhlig
reviewed by: M. Erol Sezer
MaRDI profile type: Publication
full work available at URL: 0024-3795(87)90217-5
OpenAlex ID: W1998614804
cites work: Q4754499
cites work: Q4050723
cites work: Solvability of the matric equation AX=B
cites work: Q3859620
description: scientific article; zbMATH DE number 3989561