Analysis and design of linear periodic control systems are closely related to the periodic matrix equations.The biconjugate residual method(BCR for short)have been introduced by Vespucci and Broyden for efficiently so...Analysis and design of linear periodic control systems are closely related to the periodic matrix equations.The biconjugate residual method(BCR for short)have been introduced by Vespucci and Broyden for efficiently solving linear systems Aα=b.The objective of this paper is to provide one new iterative algorithm based on BCR method to find the symmetric periodic solutions of linear periodic matrix equations.This kind of periodic matrix equations has not been dealt with yet.This iterative method is guaranteed to converge in a finite number of steps in the absence of round-off errors.Some numerical results are performed to illustrate the efficiency and feasibility of new method.展开更多
Minor self conjugate (msc) and skewpositive semidefinite (ssd) solutions to the system of matrix equations over skew fields [A mn X nn =A mn ,B sn X nn =O sn ] are considered. Necessary and su...Minor self conjugate (msc) and skewpositive semidefinite (ssd) solutions to the system of matrix equations over skew fields [A mn X nn =A mn ,B sn X nn =O sn ] are considered. Necessary and sufficient conditions for the existence of and the expressions for the msc solutions and the ssd solutions are obtained for the system.展开更多
In this article, the generalized reflexive solution of matrix equations （AX = B, XC = D） is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solv...In this article, the generalized reflexive solution of matrix equations （AX = B, XC = D） is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved.展开更多
Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew...Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew field. The representations of such the solutions of the system are also derived.展开更多
We derive necessary and sufficient conditions for the existence and an expression of the （anti）reflexive solution with respect to the nontrivial generalized reflection matrix P to the system of complex matrix equati...We derive necessary and sufficient conditions for the existence and an expression of the （anti）reflexive solution with respect to the nontrivial generalized reflection matrix P to the system of complex matrix equations AX = B and XC = D. The explicit solutions of the approximation problem min x∈Ф ｜｜X - E｜｜F was given, where E is a given complex matrix and Ф is the set of all reflexive （or antireflexive） solutions of the system mentioned above, and ｜｜·｜｜ is the Frobenius norm. Furthermore, it was pointed that some results in a recent paper are special cases of this paper.展开更多
An AOR(Accelerated Over-Relaxation)iterative method is suggested by introducing one more parameter than SOR(Successive Over-Relaxation)method for solving coupled Lyapunov matrix equations(CLMEs)that come from continuo...An AOR(Accelerated Over-Relaxation)iterative method is suggested by introducing one more parameter than SOR(Successive Over-Relaxation)method for solving coupled Lyapunov matrix equations(CLMEs)that come from continuous-time Markovian jump linear systems.The proposed algorithm improves the convergence rate,which can be seen from the given illustrative examples.The comprehensive theoretical analysis of convergence and optimal parameter needs further investigation.展开更多
In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new re...In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.展开更多
In this paper, an improved gradient iterative （GI） algorithm for solving the Lyapunov matrix equations is studied. Convergence of the improved method for any initial value is proved with some conditions. Compared wi...In this paper, an improved gradient iterative （GI） algorithm for solving the Lyapunov matrix equations is studied. Convergence of the improved method for any initial value is proved with some conditions. Compared with the GI algorithm, the improved algorithm reduces computational cost and storage. Finally, the algorithm is tested with GI several numerical examples.展开更多
Let P∈C^(n×n)be a Hermitian and{k+1}-potent matrix,i.e.,P^(k+1)=P=P^(*),where(·)^(*)stands for the conjugate transpose of a matrix.A matrix X∈C^(n×n)is called{P,k+1}-reflexive(anti-reflexive)if PXP=X(...Let P∈C^(n×n)be a Hermitian and{k+1}-potent matrix,i.e.,P^(k+1)=P=P^(*),where(·)^(*)stands for the conjugate transpose of a matrix.A matrix X∈C^(n×n)is called{P,k+1}-reflexive(anti-reflexive)if PXP=X(P XP=-X).The system of matrix equations AX=C,XB=D subject to{P,k+1}-reflexive and anti-reflexive constraints are studied by converting into two simpler cases:k=1 and k=2,the least squares solution and the associated optimal approximation problem are also considered.展开更多
The solution of two combined generalized Sylvester matrix equations is studied. It is first shown that the two combined generalized Sylvester matrix equations can be converted into a normal Sylvester matrix equation t...The solution of two combined generalized Sylvester matrix equations is studied. It is first shown that the two combined generalized Sylvester matrix equations can be converted into a normal Sylvester matrix equation through extension, and then with the help of a result for solution to normal Sylvester matrix equations, the complete solution to the two combined generalized Sylvester matrix equations is derived. A demonstrative example shows the effect of the proposed approach.展开更多
The matrix equations (AX, XBH)=(C, DH) have been widely used in structural design, parameter identification, linear optimal control, and so on. But few researches studied the reflexive solutions. A new approach for th...The matrix equations (AX, XBH)=(C, DH) have been widely used in structural design, parameter identification, linear optimal control, and so on. But few researches studied the reflexive solutions. A new approach for the reflexive solutions to the matrix equations was proposed. By applying the canonical correlation decomposition (CCD) of matrix pairs, the necessary and sufficient conditions for the existence and the general expression for the reflexive solutions of the matrix equations (AX, XBH)=(C, DH) were established. In addition, by using the methods of space decomposition, the expression of the optimal approximation solution to a given matrix was derived.展开更多
Linear matrix equations are encountered in many systems and control applications.In this paper,we consider the general coupled matrix equations（including the generalized coupled Sylvester matrix equations as a specia...Linear matrix equations are encountered in many systems and control applications.In this paper,we consider the general coupled matrix equations（including the generalized coupled Sylvester matrix equations as a special case）l t=1EstYtFst = Gs,s = 1,2,···,l over the generalized reflexive matrix group（Y1,Y2,···,Yl）.We derive an efcient gradient-iterative（GI） algorithm for fnding the generalized reflexive solution group of the general coupled matrix equations.Convergence analysis indicates that the algorithm always converges to the generalized reflexive solution group for any initial generalized reflexive matrix group（Y1（1）,Y2（1）,···,Yl（1））.Finally,numerical results are presented to test and illustrate the performance of the algorithm in terms of convergence,accuracy as well as the efciency.展开更多
This paper presents a gradient based iterative algorithm for Sylvester-conjugate matrix equations with a unique solution. By introducing a relaxation parameter and applying the hierarchical identification principle, a...This paper presents a gradient based iterative algorithm for Sylvester-conjugate matrix equations with a unique solution. By introducing a relaxation parameter and applying the hierarchical identification principle, an iterative algorithm is constructed to solve Sylvester matrix equations. By applying a real representation of a complex matrix as a tool and using some properties of the real representation, convergence analysis indicates that the iterative solutions converge to the exact solutions for any initial values under certain assumptions.Numerical examples are given to illustrate the efficiency of the proposed approach.展开更多
Dehghan and Hajarian, [4], investigated the matrix equations A^TXB＋B^TX^TA = C and A^TXB ＋ B^TXA = C providing inequalities for the determinant of the solutions of these equations. In the same paper, the authors pre...Dehghan and Hajarian, [4], investigated the matrix equations A^TXB＋B^TX^TA = C and A^TXB ＋ B^TXA = C providing inequalities for the determinant of the solutions of these equations. In the same paper, the authors presented a lower bound for the product of the eigenvalues of the solutions to these matrix equations. Inspired by their work, we give some generalizations of Dehghan and Hajarian results. Using the theory of the numerical ranges, we present an inequality involving the trace of C when A, B, X are normal matrices satisfying A^T B = BA^T.展开更多
The bi-conjugate gradients(Bi-CG)and bi-conjugate residual(Bi-CR)methods are powerful tools for solving nonsymmetric linear systems Ax=b.By using Kronecker product and vectorization operator,this paper develops the Bi...The bi-conjugate gradients(Bi-CG)and bi-conjugate residual(Bi-CR)methods are powerful tools for solving nonsymmetric linear systems Ax=b.By using Kronecker product and vectorization operator,this paper develops the Bi-CG and Bi-CR methods for the solution of the generalized Sylvester-transpose matrix equationp i=1(Ai X Bi+Ci XTDi)=E(including Lyapunov,Sylvester and Sylvester-transpose matrix equations as special cases).Numerical results validate that the proposed algorithms are much more efcient than some existing algorithms.展开更多
In this paper,we investigate the{P,Q,k+1}-reflexive and anti-reflexive solutions to the system of matrix equations AX=C,XB=D and AXB=E.We present the necessary and sufficient conditions for the system men-tioned above...In this paper,we investigate the{P,Q,k+1}-reflexive and anti-reflexive solutions to the system of matrix equations AX=C,XB=D and AXB=E.We present the necessary and sufficient conditions for the system men-tioned above to have the{P,Q,k+1}-reflexive and anti-reflexive solutions.We also obtain the expressions of such solutions to the system by the singular value decomposition.Moreover,we consider the least squares{P,Q,k+1}-reflexive and anti-reflexive solutions to the system.Finally,we give an algorithm to illustrate the results of this paper.展开更多
A class of formulas for converting linear matrix mappings into conventional linear mappings are presented. Using them, an easily computable numerical method for complete parameterized solutions of the Sylvester matrix...A class of formulas for converting linear matrix mappings into conventional linear mappings are presented. Using them, an easily computable numerical method for complete parameterized solutions of the Sylvester matrix equation AX - EXF = BY and its dual equation XA - FXE = YC are provided. It is also shown that the results obtained can be used easily for observer design. The method proposed in this paper is universally applicable to linear matrix equations.展开更多
In this paper, an iterative algorithm is presented to solve the Sylvester and Lyapunov matrix equations. By this iterative algorithm, for any initial matrix X1, a solution X＊ can be obtained within finite iteration s...In this paper, an iterative algorithm is presented to solve the Sylvester and Lyapunov matrix equations. By this iterative algorithm, for any initial matrix X1, a solution X＊ can be obtained within finite iteration steps in the absence of roundoff errors. Some examples illustrate that this algorithm is very efficient and better than that of [ 1 ] and [2].展开更多
In this paper,the quaternion matrix equations XF-AX=BY and XF-A=BY are investigated.For convenience,they were called generalized Sylvesterquaternion matrix equation and generalized Sylvester-j-conjugate quaternion mat...In this paper,the quaternion matrix equations XF-AX=BY and XF-A=BY are investigated.For convenience,they were called generalized Sylvesterquaternion matrix equation and generalized Sylvester-j-conjugate quaternion matrix equation,which include the Sylvester matrix equation and Lyapunov matrix equation as special cases.By applying of Kronecker map and complex representation of a quaternion matrix,the sufficient conditions to compute the solution can be given and the expressions of the explicit solutions to the above two quaternion matrix equations XF-AX=BY and XF-A=BY are also obtained.By the established expressions,it is easy to compute the solution of the quaternion matrix equation in the above two forms.In addition,two practical algorithms for these two quaternion matrix equations are give.One is complex representation matrix method and the other is a direct algorithm by the given expression.Furthermore,two illustrative examples are proposed to show the efficiency of the given method.展开更多
Quadratic matrix equations arise in many elds of scienti c computing and engineering applications.In this paper,we consider a class of quadratic matrix equations.Under a certain condition,we rst prove the existence of...Quadratic matrix equations arise in many elds of scienti c computing and engineering applications.In this paper,we consider a class of quadratic matrix equations.Under a certain condition,we rst prove the existence of minimal nonnegative solution for this quadratic matrix equation,and then propose some numerical methods for solving it.Convergence analysis and numerical examples are given to verify the theories and the numerical methods of this paper.展开更多
基金Supported by NSFC (No.12371378)NSF of Fujian Province (Nos.2024J01980,2023J01955)。
文摘Analysis and design of linear periodic control systems are closely related to the periodic matrix equations.The biconjugate residual method(BCR for short)have been introduced by Vespucci and Broyden for efficiently solving linear systems Aα=b.The objective of this paper is to provide one new iterative algorithm based on BCR method to find the symmetric periodic solutions of linear periodic matrix equations.This kind of periodic matrix equations has not been dealt with yet.This iterative method is guaranteed to converge in a finite number of steps in the absence of round-off errors.Some numerical results are performed to illustrate the efficiency and feasibility of new method.
文摘Minor self conjugate (msc) and skewpositive semidefinite (ssd) solutions to the system of matrix equations over skew fields [A mn X nn =A mn ,B sn X nn =O sn ] are considered. Necessary and sufficient conditions for the existence of and the expressions for the msc solutions and the ssd solutions are obtained for the system.
基金supported by National Natural Science Foundation of China (10571047)and by Scientific Research Fund of Hunan Provincial Education Department of China Grant(06C235)+1 种基金by Central South University of Forestry and Technology (06Y017)by Specialized Research Fund for the Doctoral Program of Higher Education (20060532014)
文摘In this article, the generalized reflexive solution of matrix equations （AX = B, XC = D） is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved.
基金Supported by the National Natural Science Foundation of China(10471085)
文摘Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew field. The representations of such the solutions of the system are also derived.
基金supported by the National Natural Science Foundation of China （Grant No.60672160）
文摘We derive necessary and sufficient conditions for the existence and an expression of the （anti）reflexive solution with respect to the nontrivial generalized reflection matrix P to the system of complex matrix equations AX = B and XC = D. The explicit solutions of the approximation problem min x∈Ф ｜｜X - E｜｜F was given, where E is a given complex matrix and Ф is the set of all reflexive （or antireflexive） solutions of the system mentioned above, and ｜｜·｜｜ is the Frobenius norm. Furthermore, it was pointed that some results in a recent paper are special cases of this paper.
基金Supported by Key Scientific Research Project of Colleges and Universities in Henan Province of China(Grant No.20B110012)。
文摘An AOR(Accelerated Over-Relaxation)iterative method is suggested by introducing one more parameter than SOR(Successive Over-Relaxation)method for solving coupled Lyapunov matrix equations(CLMEs)that come from continuous-time Markovian jump linear systems.The proposed algorithm improves the convergence rate,which can be seen from the given illustrative examples.The comprehensive theoretical analysis of convergence and optimal parameter needs further investigation.
基金Project supported by the National Natural Science Foundation of China （Grant No.60672160）
文摘In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.
基金Project supported by the National Natural Science Foundation of China （Grant No.10271074）, and the Special Funds for Major Specialities of Shanghai Education Commission （Grant No.J50101）
文摘In this paper, an improved gradient iterative （GI） algorithm for solving the Lyapunov matrix equations is studied. Convergence of the improved method for any initial value is proved with some conditions. Compared with the GI algorithm, the improved algorithm reduces computational cost and storage. Finally, the algorithm is tested with GI several numerical examples.
基金Supported by the Education Department Foundation of Hebei Province(QN2015218)Supported by the Natural Science Foundation of Hebei Province(A2015403050)
文摘Let P∈C^(n×n)be a Hermitian and{k+1}-potent matrix,i.e.,P^(k+1)=P=P^(*),where(·)^(*)stands for the conjugate transpose of a matrix.A matrix X∈C^(n×n)is called{P,k+1}-reflexive(anti-reflexive)if PXP=X(P XP=-X).The system of matrix equations AX=C,XB=D subject to{P,k+1}-reflexive and anti-reflexive constraints are studied by converting into two simpler cases:k=1 and k=2,the least squares solution and the associated optimal approximation problem are also considered.
基金This work was partially supported by the Chinese Outstanding Youth Foundation (No. 69925308).
文摘The solution of two combined generalized Sylvester matrix equations is studied. It is first shown that the two combined generalized Sylvester matrix equations can be converted into a normal Sylvester matrix equation through extension, and then with the help of a result for solution to normal Sylvester matrix equations, the complete solution to the two combined generalized Sylvester matrix equations is derived. A demonstrative example shows the effect of the proposed approach.
基金National Natural Science Foundation of China ( No. 60875007)
文摘The matrix equations (AX, XBH)=(C, DH) have been widely used in structural design, parameter identification, linear optimal control, and so on. But few researches studied the reflexive solutions. A new approach for the reflexive solutions to the matrix equations was proposed. By applying the canonical correlation decomposition (CCD) of matrix pairs, the necessary and sufficient conditions for the existence and the general expression for the reflexive solutions of the matrix equations (AX, XBH)=(C, DH) were established. In addition, by using the methods of space decomposition, the expression of the optimal approximation solution to a given matrix was derived.
文摘Linear matrix equations are encountered in many systems and control applications.In this paper,we consider the general coupled matrix equations（including the generalized coupled Sylvester matrix equations as a special case）l t=1EstYtFst = Gs,s = 1,2,···,l over the generalized reflexive matrix group（Y1,Y2,···,Yl）.We derive an efcient gradient-iterative（GI） algorithm for fnding the generalized reflexive solution group of the general coupled matrix equations.Convergence analysis indicates that the algorithm always converges to the generalized reflexive solution group for any initial generalized reflexive matrix group（Y1（1）,Y2（1）,···,Yl（1））.Finally,numerical results are presented to test and illustrate the performance of the algorithm in terms of convergence,accuracy as well as the efciency.
基金Supported by the National Natural Science Foundation of China(Grant No.11071033)
文摘This paper presents a gradient based iterative algorithm for Sylvester-conjugate matrix equations with a unique solution. By introducing a relaxation parameter and applying the hierarchical identification principle, an iterative algorithm is constructed to solve Sylvester matrix equations. By applying a real representation of a complex matrix as a tool and using some properties of the real representation, convergence analysis indicates that the iterative solutions converge to the exact solutions for any initial values under certain assumptions.Numerical examples are given to illustrate the efficiency of the proposed approach.
基金partially supported by FCT(Portugal)with national funds through Centro de Matemática da Universidade de Trás-os-Montes e Alto Douro(PEst-OE/MAT/UI4080/2014)
文摘Dehghan and Hajarian, [4], investigated the matrix equations A^TXB＋B^TX^TA = C and A^TXB ＋ B^TXA = C providing inequalities for the determinant of the solutions of these equations. In the same paper, the authors presented a lower bound for the product of the eigenvalues of the solutions to these matrix equations. Inspired by their work, we give some generalizations of Dehghan and Hajarian results. Using the theory of the numerical ranges, we present an inequality involving the trace of C when A, B, X are normal matrices satisfying A^T B = BA^T.
文摘The bi-conjugate gradients(Bi-CG)and bi-conjugate residual(Bi-CR)methods are powerful tools for solving nonsymmetric linear systems Ax=b.By using Kronecker product and vectorization operator,this paper develops the Bi-CG and Bi-CR methods for the solution of the generalized Sylvester-transpose matrix equationp i=1(Ai X Bi+Ci XTDi)=E(including Lyapunov,Sylvester and Sylvester-transpose matrix equations as special cases).Numerical results validate that the proposed algorithms are much more efcient than some existing algorithms.
基金supported by the National Natural Science Foundation of China(11571220)
文摘In this paper,we investigate the{P,Q,k+1}-reflexive and anti-reflexive solutions to the system of matrix equations AX=C,XB=D and AXB=E.We present the necessary and sufficient conditions for the system men-tioned above to have the{P,Q,k+1}-reflexive and anti-reflexive solutions.We also obtain the expressions of such solutions to the system by the singular value decomposition.Moreover,we consider the least squares{P,Q,k+1}-reflexive and anti-reflexive solutions to the system.Finally,we give an algorithm to illustrate the results of this paper.
基金supported by National Natural Science Foundation of China (No. 60736022, No. 60821091)
文摘A class of formulas for converting linear matrix mappings into conventional linear mappings are presented. Using them, an easily computable numerical method for complete parameterized solutions of the Sylvester matrix equation AX - EXF = BY and its dual equation XA - FXE = YC are provided. It is also shown that the results obtained can be used easily for observer design. The method proposed in this paper is universally applicable to linear matrix equations.
基金supported by the National Natural Science Foundation of China (No.10771073)
文摘In this paper, an iterative algorithm is presented to solve the Sylvester and Lyapunov matrix equations. By this iterative algorithm, for any initial matrix X1, a solution X＊ can be obtained within finite iteration steps in the absence of roundoff errors. Some examples illustrate that this algorithm is very efficient and better than that of [ 1 ] and [2].
基金This project is granted financial support from NSFC (11071079)NSFC (10901056)+2 种基金Shanghai Science and Technology Commission Venus (11QA1402200)Ningbo Natural Science Foundation (2010A610097)the Fundamental Research Funds for the Central Universities and Zhejiang Natural Science Foundation (Y6110043)
文摘In this paper,the quaternion matrix equations XF-AX=BY and XF-A=BY are investigated.For convenience,they were called generalized Sylvesterquaternion matrix equation and generalized Sylvester-j-conjugate quaternion matrix equation,which include the Sylvester matrix equation and Lyapunov matrix equation as special cases.By applying of Kronecker map and complex representation of a quaternion matrix,the sufficient conditions to compute the solution can be given and the expressions of the explicit solutions to the above two quaternion matrix equations XF-AX=BY and XF-A=BY are also obtained.By the established expressions,it is easy to compute the solution of the quaternion matrix equation in the above two forms.In addition,two practical algorithms for these two quaternion matrix equations are give.One is complex representation matrix method and the other is a direct algorithm by the given expression.Furthermore,two illustrative examples are proposed to show the efficiency of the given method.
基金Supported by the National Natural Science Foundation of China(12001395)the special fund for Science and Technology Innovation Teams of Shanxi Province(202204051002018)+1 种基金Research Project Supported by Shanxi Scholarship Council of China(2022-169)Graduate Education Innovation Project of Taiyuan Normal University(SYYJSYC-2314)。
文摘Quadratic matrix equations arise in many elds of scienti c computing and engineering applications.In this paper,we consider a class of quadratic matrix equations.Under a certain condition,we rst prove the existence of minimal nonnegative solution for this quadratic matrix equation,and then propose some numerical methods for solving it.Convergence analysis and numerical examples are given to verify the theories and the numerical methods of this paper.
