scholarly journals Explicit formulas and determinantal representation for η-skew-Hermitian solution to a system of quaternion matrix equations

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2601-2627
Author(s):  
Abdur Rehman ◽  
Ivan Kyrchei ◽  
Ilyas Ali ◽  
Muhammad Akram ◽  
Abdul Shakoor
Keyword(s):  
Sufficient Conditions ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Explicit Formulas ◽  
Determinantal Representation ◽  
System Of Matrix Equations ◽  
Representation Formulas ◽  
Special Cases ◽  
Hermitian Solution ◽  
Necessary And Sufficient

Some necessary and sufficient conditions for the existence of the ?-skew-Hermitian solution quaternion matrix equations the system of matrix equations with ?-skew-Hermicity, A1X = C1, XB1 = C2, A2Y = C3, YB2 = C4, X = -X?*; Y=-Y?*, A3XA?*3 + B3YB?*3=C5, are established in this paper by using rank equalities of the coefficient matrices. The general solutions to the system and its special cases are provided when they are consistent. Within the framework of the theory of noncommutative row-column determinants, we also give determinantal representation formulas of finding their exact solutions that are analogs of Cramer?s rule. A numerical example is also given to demonstrate the main results.

scholarly journals The General Solution of Quaternion Matrix Equation Having η-Skew-Hermicity and Its Cramer’s Rule

2019 ◽  
Vol 2019 ◽  
pp. 1-25 ◽  
Author(s):  
Abdur Rehman ◽  
Ivan Kyrchei ◽  
Ilyas Ali ◽  
Muhammad Akram ◽  
Abdul Shakoor
Keyword(s):  
General Solution ◽  
Sufficient Conditions ◽  
Quaternion Matrix ◽  
Skew Field ◽  
Determinantal Representation ◽  
Representation Formulas ◽  
Quaternion Matrix Equation ◽  
Hermitian Solution ◽  
Cramer’S Rule ◽  
Necessary And Sufficient

We determine some necessary and sufficient conditions for the existence of the η-skew-Hermitian solution to the following system AX-(AX)η⁎+BYBη⁎+CZCη⁎=D,Y=-Yη⁎,Z=-Zη⁎ over the quaternion skew field and provide an explicit expression of its general solution. Within the framework of the theory of quaternion row-column noncommutative determinants, we derive its explicit determinantal representation formulas that are an analog of Cramer’s rule. A numerical example is also provided to establish the main result.

scholarly journals Ranks of a Constrained Hermitian Matrix Expression with Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Shao-Wen Yu
Keyword(s):  
Hermitian Matrix ◽  
Schur Complement ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Necessary And Sufficient Condition ◽  
Common Solution ◽  
System Of Matrix Equations ◽  
Hermitian Solution ◽  
Matrix Expression ◽  
Necessary And Sufficient

We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expressionC4−A4XA4∗whereXis a Hermitian solution to quaternion matrix equationsA1X=C1,XB1=C2, andA3XA3*=C3. As applications, we give a new necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equationsA1X=C1,XB1=C2,A3XA3*=C3, andA4XA4*=C4, which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complementC4−A4A3~A4∗with respect to a Hermitian g-inverseA3~ofA3, which is a common solution to quaternion matrix equationsA1X=C1andXB1=C2, are also considered.

scholarly journals Explicit Determinantal Representation Formulas ofW-Weighted Drazin Inverse Solutions of Some Matrix Equations over the Quaternion Skew Field

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
Ivan I. Kyrchei
Keyword(s):  
Drazin Inverse ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Explicit Formulas ◽  
Skew Field ◽  
Determinantal Representation ◽  
Representation Formulas ◽  
Inverse Solutions ◽  
Cramer’S Rule ◽  
Weighted Drazin Inverse

By using determinantal representations of theW-weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of theW-weighted Drazin inverse solutions (analogs of Cramer’s rule) of the quaternion matrix equationsWAWX=D,XWBW=D, andW1AW1XW2BW2=D.

A System of Matrix Equations over the Quaternion Algebra with Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 233-253 ◽  
Author(s):  
Xiangrong Nie ◽  
Qingwen Wang ◽  
Yang Zhang
Keyword(s):  
General Solution ◽  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Electronic Networks ◽  
Consistency Conditions ◽  
Numerical Example ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient

We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations [Formula: see text] and [Formula: see text] over the quaternion algebra ℍ, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations [Formula: see text] over ℍ to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied.

Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications

2009 ◽  
Vol 16 (02) ◽  
pp. 293-308 ◽  
Author(s):  
Qingwen Wang ◽  
Guangjing Song ◽  
Xin Liu
Keyword(s):  
Division Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Common Solution ◽  
Special Cases ◽  
The Common ◽  
Necessary And Sufficient ◽  
Linear Matrix Equations

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A1X = C1, XB2 = C2, A3XB3 = C3 and A4XB4 = C4 over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.

scholarly journals Some quaternion matrix equations involving Φ-Hermicity

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5097-5112 ◽  
Author(s):  
Zhuo-Heng He
Keyword(s):  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Matrix Decomposition ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Existence Of A Solution ◽  
Quaternion Matrices ◽  
Real Quaternion ◽  
Necessary And Sufficient ◽  
The Given

Let H be the real quaternion algebra and Hmxn denote the set of all m x n matrices over H. For A ? Hm x n, we denote by A? the n x m matrix obtained by applying ? entrywise to the transposed matrix At, where ? is a nonstandard involution of H. A ? Hnxn is said to be ?-Hermitian if A = A?. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number (A,B,C,D), where A is ?-Hermitian, and B,C,D are general matrices. Using this simultaneous matrix decomposition, we derive necessary and sufficient conditions for the existence of a solution to some real quaternion matrix equations involving ?-Hermicity in terms of ranks of the given real quaternion matrices. We also present the general solutions to these real quaternion matrix equations when they are solvable. Finally some numerical examples are presented to illustrate the results of this paper.

scholarly journals Pure PSVD approach to Sylvester-type quaternion matrix equations

2019 ◽  
Vol 35 ◽  
pp. 266-284 ◽  
Author(s):  
Zhuo-Heng He
Keyword(s):  
General Solution ◽  
Sufficient Conditions ◽  
Singular Value ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Existence Of A Solution ◽  
Quaternion Matrices ◽  
Pure Product ◽  
Value Decomposition ◽  
Necessary And Sufficient

In this paper, the pure product singular value decomposition (PSVD) for four quaternion matrices is given. The system of coupled Sylvester-type quaternion matrix equations with five unknowns $X_{i}A_{i}-B_{i}X_{i+1}=C_{i}$ is considered by using the PSVD approach, where $A_{i},B_{i},$ and $C_{i}$ are given quaternion matrices of compatible sizes $(i=1,2,3,4)$. Some necessary and sufficient conditions for the existence of a solution to this system are derived. Moreover, the general solution to this system is presented when it is solvable.

scholarly journals On Hermitian Solutions of the Generalized Quaternion Matrix Equation A X B + C X   D = E

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Yong Tian ◽  
Xin Liu ◽  
Shi-Fang Yuan
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Quaternion Matrix ◽  
Real Representation ◽  
Quaternion Algebras ◽  
The Matrix ◽  
Hermitian Solution ◽  
Necessary And Sufficient ◽  
Generalized Quaternion ◽  
Algebraic Technique

The paper deals with the matrix equation A X B + C X   D = E over the generalized quaternions. By the tools of the real representation of a generalized quaternion matrix, Kronecker product as well as vec-operator, the paper derives the necessary and sufficient conditions for the existence of a Hermitian solution and gives the explicit general expression of the solution when it is solvable and provides a numerical example to test our results. The paper proposes a unificated algebraic technique for finding Hermitian solutions to the mentioned matrix equation over the generalized quaternions, which includes many important quaternion algebras, such as the Hamilton quaternions and the split quaternions.

scholarly journals Constrained Solutions of a System of Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Qing-Wen Wang ◽  
Juan Yu
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
System Of Matrix Equations ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Constrained Solutions

We derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equationsAX=BandXC=D, respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skew-symmetric orthogonal solutions are respectively given. As an auxiliary, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable.

scholarly journals Real Representation Approach to Quaternion Matrix Equation Involving ϕ-Hermicity

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Xin Liu ◽  
Huajun Huang ◽  
Zhuo-Heng He
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Quaternion Matrix ◽  
Real Representation ◽  
Quaternion Matrix Equation ◽  
The Matrix ◽  
Hermitian Solution ◽  
Necessary And Sufficient

For a quaternion matrix A, we denote by Aϕ the matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a nonstandard involution of quaternions. A is said to be ϕ-Hermitian or ϕ-skew-Hermitian if A=Aϕ or A=−Aϕ, respectively. In this paper, we give a complete characterization of the nonstandard involutions ϕ of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution to the quaternion matrix equation AX=B. Moreover, we give solutions of the quaternion equation when it is solvable.

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