GENERAL Φ-HERMITIAN SOLUTION TO A SYSTEM OF QUATERNION MATRIX EQUATIONS

2019 ◽  
Vol 41 (1) ◽  
pp. 35-47
Author(s):  
Mengyan Xie ◽  
Pingping Song ◽  
Zhiqing Zhang
Keyword(s):  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Solution

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 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 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.

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

2015 ◽  
Vol 30 ◽  
Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Iterative Algorithms ◽  
Iterative Approach ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Convergence Properties ◽  
Least Squares Problems

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

Least-norm and Extremal Ranks of the Least Square Solution to the Quaternion Matrix Equation AXB=C Subject to Two Equations

2014 ◽  
Vol 21 (03) ◽  
pp. 449-460 ◽  
Author(s):  
Yubao Bao
Keyword(s):  
Matrix Equation ◽  
Least Square ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Consistent System ◽  
Quaternion Matrix Equation

In this paper, we give the expression of the least square solution of the linear quaternion matrix equation AXB=C subject to a consistent system of quaternion matrix equations D1X=F1, XE2=F2, and derive the maximal and minimal ranks and the least-norm of the above mentioned solution. The finding of this paper extends some known results in the literature.

The Real Solutions to a System of Quaternion Matrix Equations with Applications

2009 ◽  
Vol 37 (6) ◽  
pp. 2060-2079 ◽  
Author(s):  
Qing-Wen Wang ◽  
Shao-Wen Yu ◽  
Qin Zhang
Keyword(s):  
Matrix Equations ◽  
Quaternion Matrix ◽  
The Real ◽  
Real Solutions
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