# 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 Journals
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations