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.

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.

Solution to a Class of Matrix Equations with k-Involutary Symmetrices

2012 ◽  
Vol 457-458 ◽  
pp. 799-803 ◽  
Author(s):  
Mao Lin Liang ◽  
Li Fang Dai
Keyword(s):  
General Solution ◽  
Symmetric Matrix ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Matrix Equations

In this paper, we investigate the solvability of matrix equations with -involutary symmetric matrix , the general solution of which is obtained when it is solvable. Meantime, the associated optimal approximation problem for some given matrix is also considered under some particular hypothesis.

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 Solvability Conditions and General Solution of a System of Matrix Equations Involving η-Skew-Hermitian Quaternion Matrices

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1825
Author(s):  
Abdur Rehman ◽  
Israr Ali Khan ◽  
Rukhshanda Anjum ◽  
Iftikhar Hussain
Keyword(s):  
General Solution ◽  
Matrix Equations ◽  
Solvability Conditions ◽  
Numerical Example ◽  
Quaternion Matrices ◽  
System Of Matrix Equations ◽  
Special Cases

In this article, we study the solvability conditions and the general solution of a system of matrix equations involving η-skew-Hermitian quaternion matrices. Several special cases of this system are discussed, and we recover some well-known results in the literature. An algorithm and a numerical example for the validation of our main result are also provided.

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.

scholarly journals A class of constrained inverse eigenvalue problem and associated approximation problem for symmetrizable matrices

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 1903-1909
Author(s):  
Xiangyang Peng ◽  
Wei Liu ◽  
Jinrong Shen
Keyword(s):  
Eigenvalue Problem ◽  
Symmetric Matrix ◽  
Approximation Problem ◽  
Inverse Eigenvalue Problem ◽  
Optimal Approximation ◽  
Original Matrix ◽  
Approximation Solution ◽  
Solvability Conditions ◽  
Inverse Eigenvalue ◽  
Symmetrizable Matrices

The real symmetric matrix is widely applied in various fields, transforming non-symmetric matrix to symmetric matrix becomes very important for solving the problems associated with the original matrix. In this paper, we consider the constrained inverse eigenvalue problem for symmetrizable matrices, and obtain the solvability conditions and the general expression of the solutions. Moreover, we consider the corresponding optimal approximation problem, obtain the explicit expressions of the optimal approximation solution and the minimum norm solution, and give the algorithm and corresponding computational example.

scholarly journals The least-squares solutions of the matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ and its optimal approximation

2021 ◽  
Vol 7 (3) ◽  
pp. 3680-3691
Author(s):  
Huiting Zhang ◽  
◽  
Yuying Yuan ◽  
Sisi Li ◽  
Yongxin Yuan ◽  
...  
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Canonical Correlation ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Linear Matrix ◽  
General Representation ◽  
Linear Matrix Equation ◽  
The Matrix

<abstract><p>In this paper, the least-squares solutions to the linear matrix equation $ A^{\ast}XB+B^{\ast}X^{\ast}A = D $ are discussed. By using the canonical correlation decomposition (CCD) of a pair of matrices, the general representation of the least-squares solutions to the matrix equation is derived. Moreover, the expression of the solution to the corresponding weighted optimal approximation problem is obtained.</p></abstract>

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.

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