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 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 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 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 Determinantal Representations of General and (Skew-)Hermitian Solutions to the Generalized Sylvester-Type Quaternion Matrix Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Ivan I. Kyrchei
Keyword(s):  
Matrix Equation ◽  
Quaternion Matrix ◽  
Sylvester Matrix Equation ◽  
Skew Field ◽  
Determinantal Representation ◽  
Sylvester Matrix ◽  
Generalized Sylvester Matrix Equation ◽  
Representation Formulas ◽  
Quaternion Matrix Equation

In this paper, we derive explicit determinantal representation formulas of general, Hermitian, and skew-Hermitian solutions to the generalized Sylvester matrix equation involving ⁎-Hermicity AXA⁎+BYB⁎=C over the quaternion skew field within the framework of the theory of noncommutative column-row determinants.

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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