Cramer's rules for Sylvester quaternion matrix equation and its special cases
type: Item
instance of: scholarly article
title: Cramer's rules for Sylvester quaternion matrix equation and its special cases
zbMATH Open document ID: 1404.15013
author: Ivan I. Kyrchei
published in: Advances in Applied Clifford Algebras
publication date: 2018-10-25T00:00:00Z
review text: The author considers the problem of solving the classical Sylvester-type linear equation \(A_{1}XB_{1}+A_{2}XB_{2}=C\) for matrices over the ring \(\mathbb{H}\) of quaternions, where \(X\) is an unknown (in general rectangular) matrix. This is a classical problem which has been thoroughly studied over \(\mathbb{R}\) and \(\mathbb{C}\) and more recently over the quaternions. In [Linear Algebra Appl. 431, No. 12, 2291--2303 (2009; Zbl 1180.15019)], \textit{Q.-W. Wang} et al. showed how to express the solutions to the above equation in terms of generalized inverses and recently the current author [Linear Multilinear Algebra 59, No. 4, 413--431 (2011; Zbl 1220.15007)] used a quaternionic analogue of Cramer's rule to express the Moore-Penrose inverse of a matrix over \(\mathbb{H}\). In the present paper, these earlier results are combined to give necessary and sufficient conditions for the Sylvester-type linear equation to have a solution, and to write down explicit solutions in terms of the entries of \(A_{1}\), \(A_{2}\), \(B_{1}\), \(B_{2}\) and \(C\). Particular cases such as \(AX+XB=C\) are worked out in detail.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 6965679
zbMATH Keywords: Sylvester matrix equation
zbMATH Keywords: Cramer rule
zbMATH Keywords: quaternion matrix
zbMATH Keywords: noncommutative determinant
zbMATH Keywords: Lyapunov matrix equation
MaRDI profile type: Publication
full work available at URL: s00006-018-0909-0
OpenAlex ID: W2894046110
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DOI: 10.1007/S00006-018-0909-0