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14 pages, 273 KiB

Open AccessArticle

Dual Quaternion Matrix Equation AXB = C with Applications

by Yan Chen, Qing-Wen Wang and Lv-Ming Xie

Symmetry 2024, 16(3), 287; https://doi.org/10.3390/sym16030287 - 1 Mar 2024

Cited by 29 | Viewed by 2316

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation

A X B = C

has been extensively studied. However, there is currently limited information on matrix equation [...] Read more.

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation

A X B = C

has been extensively studied. However, there is currently limited information on matrix equation

A X B = C

regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation

A X B = C

, and present the expression for the general solution when it is solvable. As an application, we derive the

ϕ

-Hermitian solutions for dual quaternion matrix equation

A X A^{ϕ} = C

, where the

ϕ

-Hermitian extends the concepts of Hermiticity and

η

-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper. Full article

(This article belongs to the Section Mathematics)

28 pages, 339 KiB

Open AccessArticle

Some Properties of the Solution to a System of Quaternion Matrix Equations

by Shao-Wen Yu, Xiao-Na Zhang, Wei-Lu Qin and Zhuo-Heng He

Axioms 2022, 11(12), 710; https://doi.org/10.3390/axioms11120710 - 8 Dec 2022

Viewed by 1468

Abstract

This paper investigates the properties of the

ϕ

-skew-Hermitian solution to the system of quaternion matrix equations involving

ϕ

-skew-Hermicity with four unknowns [...] Read more.

This paper investigates the properties of the

ϕ

-skew-Hermitian solution to the system of quaternion matrix equations involving

ϕ

-skew-Hermicity with four unknowns

A_{i} X_{i} {(A_{i})}_{ϕ} + B_{i} X_{i + 1} {(B_{i})}_{ϕ} = C_{i}, (i = 1, 2, 3), A_{4} X_{4} {(A_{4})}_{ϕ} = C_{4}

. We present the general

ϕ

-skew-Hermitian solution to this system. Moreover, we derive the

β (ϕ)

-signature bounds of the

ϕ

-skew-Hermitian solution

X_{1}

in terms of the coefficient matrices. We also give some necessary and sufficient conditions for the system to have

β

(

ϕ

)-positive semidefinite,

β

(

ϕ

)-positive definite,

β

(

ϕ

)-negative semidefinite and

β

(

ϕ

)-negative definite solutions. Full article

(This article belongs to the Special Issue Advances in Linear Algebra)

18 pages, 310 KiB

Open AccessArticle

Consistency and General Solutions to Some Sylvester-like Quaternion Matrix Equations

by Zhuo-Heng He, Jie Tian, Yun-Fan Zhao and Shao-Wen Yu

Symmetry 2022, 14(7), 1350; https://doi.org/10.3390/sym14071350 - 30 Jun 2022

Cited by 1 | Viewed by 1539

Abstract

This article makes use of simultaneous decomposition of four quaternion matrixes to investigate some Sylvester-like quaternion matrix equation systems. We present some useful necessary and sufficient conditions for the consistency of the system of quaternion matrix equations in terms of the equivalence form and block matrixes. We also derive the general solution to the system according to the partition of the coefficient matrixes. As an application of the system, we present some practical necessary and sufficient conditions for the consistency of a

ϕ

-Hermitian solution to the system of quaternion matrix equations in terms of the equivalence form and block matrixes. We also provide the general

ϕ

-Hermitian solution to the system when the equation system is consistent. Moreover, we present some numerical examples to illustrate the availability of the results of this paper. Full article

(This article belongs to the Section Mathematics)

19 pages, 285 KiB

Open AccessArticle

The Solvability of a System of Quaternion Matrix Equations Involving ϕ-Skew-Hermicity

by Zhuo-Heng He, Xiao-Na Zhang, Yun-Fan Zhao and Shao-Wen Yu

Symmetry 2022, 14(6), 1273; https://doi.org/10.3390/sym14061273 - 20 Jun 2022

Cited by 3 | Viewed by 1769

Abstract

Let

H

be the real quaternion algebra and

H^{m \times n}

denote the set of all

m \times n

matrices over

H

. For

A \in H^{m \times n},

we denote by

A_{ϕ}

the

n \times m

matrix obtained [...] Read more.

Let

H

be the real quaternion algebra and

H^{m \times n}

denote the set of all

m \times n

matrices over

H

. For

A \in H^{m \times n},

we denote by

A_{ϕ}

the

n \times m

matrix obtained by applying

ϕ

entrywise to the transposed matrix

A^{T},

where

ϕ

is a non-standard involution of

H

A \in H^{n \times n}

is said to be

ϕ

-skew-Hermicity if

A = - A_{ϕ}

. In this paper, we provide some necessary and sufficient conditions for the existence of a

ϕ

-skew-Hermitian solution to the system of quaternion matrix equations with four unknowns

A_{i} X_{i} {(A_{i})}_{ϕ} + B_{i} X_{i + 1} {(B_{i})}_{ϕ} = C_{i}, (i = 1, 2, 3), A_{4} X_{4} {(A_{4})}_{ϕ} = C_{4}

. Full article

(This article belongs to the Special Issue Selected Papers from the 2021 International Conference on Matrix Inequalities and Matrix Equations)

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