The Solution of a Generalized Sylvester Quaternion Matrix Equation and Its Application

The Solution of a Generalized Sylvester Quaternion Matrix Equation and Its Application An $$n\times n$$ n × n quaternion matrix is said to be $$\eta$$ η -Hermitian if $$A=A^{\eta {*}}$$ A = A η ∗ , where $$A^{\eta {*}}=-\eta A^{*}\eta$$ A η ∗ = - η A ∗ η , $$\eta$$ η is one of the quaternion units i, j, k, and $$A^{*}$$ A ∗ is the conjugate transpose of A. In this paper, we investigate the generalized Sylvester quaternion matrix equation \begin{aligned} A_{1}X_{1}B_{1}+A_{2}X_{2}B_{2}+A_{3}X_{3}B_{3}=C. \end{aligned} A 1 X 1 B 1 + A 2 X 2 B 2 + A 3 X 3 B 3 = C . We establish the necessary and sufficient conditions for the existence of a solution to this equation, and give an expression of the general solution to the equation when it is solvable. As an application, we derive the solvability conditions for the quaternion matrix equation \begin{aligned} A_{1}X_{1}A_{1}^{\eta *}+A_{2}X_{2}A_{2}^{\eta *}+A_{3}X_{3}A_{3} ^{\eta *}=C \end{aligned} A 1 X 1 A 1 η ∗ + A 2 X 2 A 2 η ∗ + A 3 X 3 A 3 η ∗ = C to have an $$\eta$$ η -Hermitian solution as well as an expression of the $$\eta$$ η -Hermitian solution. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Applied Clifford Algebras Springer Journals

The Solution of a Generalized Sylvester Quaternion Matrix Equation and Its Application

, Volume 27 (3) – Apr 22, 2017
20 pages

/lp/springer_journal/the-solution-of-a-generalized-sylvester-quaternion-matrix-equation-and-SJFJfsS9WU
Publisher
Springer International Publishing
Subject
Physics; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Applications of Mathematics; Physics, general
ISSN
0188-7009
eISSN
1661-4909
D.O.I.
10.1007/s00006-017-0782-2
Publisher site
See Article on Publisher Site

Abstract

An $$n\times n$$ n × n quaternion matrix is said to be $$\eta$$ η -Hermitian if $$A=A^{\eta {*}}$$ A = A η ∗ , where $$A^{\eta {*}}=-\eta A^{*}\eta$$ A η ∗ = - η A ∗ η , $$\eta$$ η is one of the quaternion units i, j, k, and $$A^{*}$$ A ∗ is the conjugate transpose of A. In this paper, we investigate the generalized Sylvester quaternion matrix equation \begin{aligned} A_{1}X_{1}B_{1}+A_{2}X_{2}B_{2}+A_{3}X_{3}B_{3}=C. \end{aligned} A 1 X 1 B 1 + A 2 X 2 B 2 + A 3 X 3 B 3 = C . We establish the necessary and sufficient conditions for the existence of a solution to this equation, and give an expression of the general solution to the equation when it is solvable. As an application, we derive the solvability conditions for the quaternion matrix equation \begin{aligned} A_{1}X_{1}A_{1}^{\eta *}+A_{2}X_{2}A_{2}^{\eta *}+A_{3}X_{3}A_{3} ^{\eta *}=C \end{aligned} A 1 X 1 A 1 η ∗ + A 2 X 2 A 2 η ∗ + A 3 X 3 A 3 η ∗ = C to have an $$\eta$$ η -Hermitian solution as well as an expression of the $$\eta$$ η -Hermitian solution.

Journal

Advances in Applied Clifford AlgebrasSpringer Journals

Published: Apr 22, 2017

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