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

The Real Solutions of Equation of in Control Theory

2010 ◽  
Vol 121-122 ◽  
pp. 911-915
Author(s):  
Xue Ting Liu
Keyword(s):  
Control Theory ◽  
Chemical Engineering ◽  
Research Field ◽  
Matrix Equations ◽  
The Real ◽  
Real Solutions ◽  
Nonsingular Solution ◽  
The Matrix ◽  
Active Research ◽  
Solutions Of Equations

. The research of matrix equations is an active research field, matrix equations have applied in many physical applications in recent years. As one of them, the equation is applied more and more extensively, such as control theory, chemistry and chemical engineering and so on. In this paper, motivated by [1], we give two discriminations about the real solutions of equation . The matrix is proved that it is a nonsingular solution of equation whenever are nonsingular solutions of equations at last.

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.

Microcomputer-Assisted Mathematics: Polynomial Equations Revisited

1986 ◽  
Vol 79 (9) ◽  
pp. 732-737
Author(s):  
Jillian C. F. Sullivan
Keyword(s):  
High School ◽  
High School Students ◽  
Numerical Methods ◽  
Polynomial Equations ◽  
School Students ◽  
The Real ◽  
Real Solutions ◽  
Quadratic Equations ◽  
Computational Difficulty

Although solving polynomial equations is important in mathematics, most high school students can solve only linear and quadratic equations. This is because the methods for solving cubic and quartic equations are difficult, and no general methods of solution are available for equations of degree higher than four. However, numerical methods can be used to approximate the real solutions of polynomial equations of any degree. Because they involve a great deal of computation they have not traditionally been taught in the schools. Now that most students have access to calculators and computers, this computational difficulty is easily overcome.

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.

Sign in / Sign up

Export Citation Format

Share Document