scholarly journals On the HermitianR-Conjugate Solution of a System of Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Chang-Zhou Dong ◽  
Qing-Wen Wang ◽  
Yu-Ping Zhang
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Approximation Problem ◽  
Matrix Equations ◽  
Complex Matrix ◽  
Solvability Conditions ◽  
Numerical Examples ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient ◽  
Conjugate Solution

LetRbe annbynnontrivial real symmetric involution matrix, that is,R=R−1=RT≠In. Ann×ncomplex matrixAis termedR-conjugate ifA¯=RAR, whereA¯denotes the conjugate ofA. We give necessary and sufficient conditions for the existence of the HermitianR-conjugate solution to the system of complex matrix equationsAX=C and XB=Dand present an expression of the HermitianR-conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares HermitianR-conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.

scholarly journals (Anti-)Hermitian Generalized (Anti-)Hamiltonian Solution to a System of Matrix Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Juan Yu ◽  
Qing-Wen Wang ◽  
Chang-Zhou Dong
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Optimal Approximation ◽  
Matrix Equations ◽  
Approximation Solution ◽  
Numerical Examples ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient ◽  
The Given

We mainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions to the system of matrix equationsAX=B,XC=Dare derived, respectively. Secondly, the optimal approximation solutionmin⁡X∈K⁡∥X^-X∥is obtained, whereKis the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of the above system andX^is the given matrix. Thirdly, the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solutions are considered. In addition, algorithms about computing the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solution and the corresponding numerical examples are presented.

scholarly journals Constrained Solutions of a System of Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Qing-Wen Wang ◽  
Juan Yu
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
System Of Matrix Equations ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Constrained Solutions

We derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equationsAX=BandXC=D, respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skew-symmetric orthogonal solutions are respectively given. As an auxiliary, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable.

scholarly journals The Hermitian -Conjugate Generalized Procrustes Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Hai-Xia Chang ◽  
Xue-Feng Duan ◽  
Qing-Wen Wang
Keyword(s):  
Sufficient Conditions ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Complex Matrix ◽  
Approximation Solution ◽  
Hermitian Conjugate ◽  
Procrustes Problem ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Conjugate Solution

We consider the Hermitian -conjugate generalized Procrustes problem to find Hermitian -conjugate matrix such that is minimum, where , , , and (, ) are given complex matrices, and and are positive integers. The expression of the solution to Hermitian -conjugate generalized Procrustes problem is derived. And the optimal approximation solution in the solution set for Hermitian -conjugate generalized Procrustes problem to a given matrix is also obtained. Furthermore, we establish necessary and sufficient conditions for the existence and the formula for Hermitian -conjugate solution to the linear system of complex matrix equations , , ( and are positive integers). The representation of the corresponding optimal approximation problem is presented. Finally, an algorithm for solving two problems above is proposed, and the numerical examples show its feasibility.

scholarly journals The{P,Q,k+1}-Reflexive Solution to System of Matrix EquationsAX=C,XB=D

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Chang-Zhou Dong ◽  
Qing-Wen Wang
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Matrix Equations ◽  
Solvability Conditions ◽  
Least Squares Solution ◽  
System Of Matrix Equations ◽  
Conjugate Transpose

LetP∈Cm×mandQ∈Cn×nbe Hermitian and{k+1}-potent matrices; that is,Pk+1=P=P⁎andQk+1=Q=Q⁎,where·⁎stands for the conjugate transpose of a matrix. A matrixX∈Cm×nis called{P,Q,k+1}-reflexive (antireflexive) ifPXQ=X (PXQ=-X). In this paper, the system of matrix equationsAX=CandXB=Dsubject to{P,Q,k+1}-reflexive and antireflexive constraints is studied by converting into two simpler cases:k=1andk=2.We give the solvability conditions and the general solution to this system; in addition, the least squares solution is derived; finally, the associated optimal approximation problem for a given matrix is considered.

A System of Matrix Equations over the Quaternion Algebra with Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 233-253 ◽  
Author(s):  
Xiangrong Nie ◽  
Qingwen Wang ◽  
Yang Zhang
Keyword(s):  
General Solution ◽  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Electronic Networks ◽  
Consistency Conditions ◽  
Numerical Example ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient

We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations [Formula: see text] and [Formula: see text] over the quaternion algebra ℍ, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations [Formula: see text] over ℍ to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied.

On Solutions of Inverse Problem for Hermitian Generalized Hamiltonian Matrices

2013 ◽  
Vol 860-863 ◽  
pp. 2727-2731
Author(s):  
Kai Fu Liang ◽  
Ming Jun Li ◽  
Ze Lin Zhu
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Design Automation ◽  
Sufficient Conditions ◽  
Matrix Equations ◽  
Complex Matrix ◽  
Linear Quadratic ◽  
Real Representation ◽  
The Matrix ◽  
Necessary And Sufficient

Hamiltonian matrices have many applications to design automation and autocontrol, in particular in the linear-quadratic autocontrol problem. This paper studies the inverse problems of generalized Hamiltonian matrices for matrix equations. By real representation of complex matrix, we give the necessary and sufficient conditions for the existence of a Hermitian generalized Hamiltonian solutions to the matrix equations, and then derive the representation of the general solutions.

Solvability conditions for mixed sylvester equations in rings

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5539-5543 ◽  
Author(s):  
Huaxi Chen ◽  
Long Wang ◽  
Qian Wang
Keyword(s):  
General Solution ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Solvability Conditions ◽  
Sylvester Matrix ◽  
Necessary And Sufficient ◽  
Sylvester Matrix Equations ◽  
Algebraic Technique

This paper has been motivated byWang and He [Q.W.Wang and Z.H. He, Solvability conditions and general solution for mixed Sylvester equations. Automatica, 49 (2013) 2713-2719] in which the authors consider some solvability conditions for mixed Sylvester matrix equations. The paper also considers the same problem in the setting of a regular ring. Using the purely algebraic technique, we present some necessary and sufficient conditions for the solvability to mixed Sylvester equations in rings.

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.

On the Centro-symmetric Solution of a System of Matrix Equations over a Regular Ring with Identity

2007 ◽  
Vol 14 (04) ◽  
pp. 555-570 ◽  
Author(s):  
Qingwen Wang ◽  
Haixia Chang ◽  
Chunyan Lin
Keyword(s):  
General Solution ◽  
Explicit Expression ◽  
Symmetric Solution ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient

In this paper, we find the centro-symmetric solution of a system of matrix equations over an arbitrary regular ring [Formula: see text] with identity. We first derive some necessary and sufficient conditions for the existence and an explicit expression of the general solution of the system of matrix equations A1X1 = C1, A2X1 = C2, A3X2 = C3, A4X2 = C4 and A5X1B5 + A6X2B6 = C5 over [Formula: see text]. By using the above results, we establish two criteria for the existence and the representation of the general centro-symmetric solution of the system of matrix equations AaX = Ca, AbX = Cb and AcXBc = Cc over the ring [Formula: see text].

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