scholarly journals Cramer’s rules for the system of quaternion matrix equations with η-Hermicity

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 24 ◽  
Author(s):  
Ivan I. Kyrchei
Keyword(s):  
General Solution ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Special Cases ◽  
Hermitian Solution ◽  
Cramer’S Rule

The system of two-sided quaternion matrix equations with η-Hermicity, A1XA1η* = C1, A2XA2η* = C2 is considered in the paper. Using noncommutative row-column determinants previously introduced by the author, determinantal representations (analogs of Cramer’s rule) of a general solution to the system are obtained. As special cases, Cramer’s rules for an η-Hermitian solution when C1 = Cη*1 and C2 = Cη*2 and for an η-skew-Hermitian solution when C1 = −Cη*1 and C2 = −Cη*2 are also explored.

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 Least-Norm of the General Solution to Some System of Quaternion Matrix Equations and Its Determinantal Representations

2019 ◽  
Vol 2019 ◽  
pp. 1-18
Author(s):  
Abdur Rehman ◽  
Ivan Kyrchei ◽  
Muhammad Akram ◽  
Ilyas Ali ◽  
Abdul Shakoor
Keyword(s):  
General Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Skew Field ◽  
Numerical Examples ◽  
Cramer’S Rule ◽  
Necessary And Sufficient

We constitute some necessary and sufficient conditions for the system A1X1=C1, X1B1=C2, A2X2=C3, X2B2=C4, A3X1B3+A4X2B4=Cc, to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.

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 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.

Cramer’s rule for a system of quaternion matrix equations with applications

2018 ◽  
Vol 336 ◽  
pp. 490-499 ◽  
Author(s):  
Guang-Jing Song ◽  
Qing-Wen Wang ◽  
Shao-Wen Yu
Keyword(s):  
Matrix Equations ◽  
Quaternion Matrix ◽  
Cramer’S Rule

scholarly journals Determinantal Representations of Solutions and Hermitian Solutions to Some System of Two-Sided Quaternion Matrix Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Ivan I. Kyrchei
Keyword(s):  
Matrix Equations ◽  
Quaternion Matrix ◽  
Cramer’S Rule

Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we derive determinantal representations (analogs of Cramer’s rule) of solutions and Hermitian solutions to the system of two-sided quaternion matrix equations A1XA1⁎=C1 and A2XA2⁎=C2. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use determinantal representations of the Moore-Penrose inverse previously obtained by the theory of row-column determinants.

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