Constraint Solution of a Classical System of Quaternion Matrix Equations and Its Cramer’s Rule

2021 ◽  
Author(s):  
Abdur Rehman ◽  
Ivan Kyrchei ◽  
Ilyas Ali ◽  
Muhammad Akram ◽  
Abdul Shakoor
Keyword(s):  
Classical System ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Cramer’S Rule

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.

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 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 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.

scholarly journals The Real and Complex Hermitian Solutions to a System of Quaternion Matrix Equations with Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Shao-Wen Yu
Keyword(s):  
Classical System ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
The Real ◽  
Necessary And Sufficient ◽  
Real Matrices

We establish necessary and sufficient conditions for the existence of and the expressions for the general real and complex Hermitian solutions to the classical system of quaternion matrix equationsA1X=C1,XB1=C2, and  A3XA3*=C3. Moreover, formulas of the maximal and minimal ranks of four real matricesX1,X2,X3, andX4in solutionX=X1+X2i+X3j+X4kto the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equationsA1X=C1,XB1=C2,A3XA3*=C3, and  A4XA4*=C4to have real and complex Hermitian solutions.

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