Based on the well-known Leverrier algorithm, a simple explicit solution to right factorization of a linear system is established. This solution is expressed by the controllability matrix of the given system and a symm...Based on the well-known Leverrier algorithm, a simple explicit solution to right factorization of a linear system is established. This solution is expressed by the controllability matrix of the given system and a symmetric operator matrix. Applications of this solution to a type of generalized Sylvester matrix equatiorls and the problem of parametric eigenstructure assignment by state feedback are investigated,and general complete parametric solutions to these two problems are deduced. These new solutions are simple, and possess desirable structural properties which render the solutions readily implementable. An example demonstrates the effect of the proposed results.展开更多
We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their...We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their solutions.As an application,we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayedmatrix functions.The obtained results are new,and they extend and improve some existing ones.Finally,an example is presented to illustrate the validity of our theoretical results.展开更多
A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provi...A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in descriptor system theory. The results proposed here are parallel to and more general than our early work about the linear matrix equation AX-XF = BY .展开更多
In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of s...In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of solving(I)is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.展开更多
In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are est...In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × （n ＋ pr）. The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair （A, B） and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.展开更多
The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For thi...The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.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 ∈ Cn×nis called{P, k + 1}-reflexive(anti-reflexive...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 ∈ Cn×nis 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
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, we introduce a Hermite operational matrix collocation method for solving higher-order linear complex differential equations in rectangular or elliptic domains. We show that based on a linear algebra the...In this paper, we introduce a Hermite operational matrix collocation method for solving higher-order linear complex differential equations in rectangular or elliptic domains. We show that based on a linear algebra theorem, the use of different polynomials such as Hermite, Bessel and Taylor in polynomial collocation methods for solving differential equations leads to an equal solution, and the difference in the numerical results arises from the difference in the coefficient matrix of final linear systems of equations. Some numerical examples will also be given.展开更多
In the paper, a class of fuzzy matrix equations AX=B where A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp m...In the paper, a class of fuzzy matrix equations AX=B where A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp matrix equations. Then the fuzzy approximate solution of the fuzzy matrix equation is obtained by solving two crisp matrix equations. The existence condition of the strong LR fuzzy solution to the fuzzy matrix equation is also discussed. Some examples are given to illustrate the proposed method. Our results enrich the fuzzy linear systems theory.展开更多
This work is concerned with the nonlinear matrix equation Xs ＋ A＊F（X）A = Q with s ≥ 1. Several sufficient and necessary conditions for the existence and uniqueness of the Hermitian positive semidefinite solution ...This work is concerned with the nonlinear matrix equation Xs ＋ A＊F（X）A = Q with s ≥ 1. Several sufficient and necessary conditions for the existence and uniqueness of the Hermitian positive semidefinite solution are derived, and perturbation bounds are presented.展开更多
In this note, the matrix equation AV ＋ BW = EVJ is considered, where E, A and B are given matrices of appropriate dimensions, J is an arbitrarily given Jordan matrix, V and W are the matrices to be determined. Firstl...In this note, the matrix equation AV ＋ BW = EVJ is considered, where E, A and B are given matrices of appropriate dimensions, J is an arbitrarily given Jordan matrix, V and W are the matrices to be determined. Firstly, a right factorization of （sE - A）^-1 B is given based on the Leverriver algorithm for descriptor systems. Then based on this factorization and a proposed parametric solution, an alternative parametric solution to this matrix equation is established in terms of the R-controllability matrix of （E, A, B）, the generalized symmetric operator and the observability matrix associated with the Jordan matrix d and a free parameter matrix. The proposed results provide great convenience for many analysis and design problems. Moreover, some equivalent forms are proposed. A numerical example is employed to illustrate the effect of the proposed approach.展开更多
An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations i...An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.展开更多
基金This work was supported bythe Chinese Outstanding Youth Foundation (No .69504002) .
文摘Based on the well-known Leverrier algorithm, a simple explicit solution to right factorization of a linear system is established. This solution is expressed by the controllability matrix of the given system and a symmetric operator matrix. Applications of this solution to a type of generalized Sylvester matrix equatiorls and the problem of parametric eigenstructure assignment by state feedback are investigated,and general complete parametric solutions to these two problems are deduced. These new solutions are simple, and possess desirable structural properties which render the solutions readily implementable. An example demonstrates the effect of the proposed results.
文摘We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay.By using new conformable delayed matrix functions and the method of variation,we obtain a representation of their solutions.As an application,we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayedmatrix functions.The obtained results are new,and they extend and improve some existing ones.Finally,an example is presented to illustrate the validity of our theoretical results.
基金supported by the Major Program of National Nat-ural Science Foundation of China (No. 60710002) Program for Changjiang Scholars and Innovative Research Team in University
文摘A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in descriptor system theory. The results proposed here are parallel to and more general than our early work about the linear matrix equation AX-XF = BY .
文摘In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of solving(I)is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.
基金This work was supported by the Chinese Outstanding Youth Foundation(No.69925308)Program for Changjiang Scholars and Innovative ResearchTeam in University.
文摘In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × （n ＋ pr）. The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair （A, B） and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.
文摘The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.
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 ∈ Cn×nis 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.
文摘In this paper, we introduce a Hermite operational matrix collocation method for solving higher-order linear complex differential equations in rectangular or elliptic domains. We show that based on a linear algebra theorem, the use of different polynomials such as Hermite, Bessel and Taylor in polynomial collocation methods for solving differential equations leads to an equal solution, and the difference in the numerical results arises from the difference in the coefficient matrix of final linear systems of equations. Some numerical examples will also be given.
文摘In the paper, a class of fuzzy matrix equations AX=B where A is an m × n crisp matrix and is an m × p arbitrary LR fuzzy numbers matrix, is investigated. We convert the fuzzy matrix equation into two crisp matrix equations. Then the fuzzy approximate solution of the fuzzy matrix equation is obtained by solving two crisp matrix equations. The existence condition of the strong LR fuzzy solution to the fuzzy matrix equation is also discussed. Some examples are given to illustrate the proposed method. Our results enrich the fuzzy linear systems theory.
基金The authors are very much indebted to the referees for their constructive and valuable comments and suggestions which greatly improved the original manuscript of this paper. This work of the first author is supported by Scholarship Award for Excellent Doctoral Student granted by East China Normal University （No.XRZZ2012021）. This work of the second author is supported by the National Natural Science Foundation of China （No. 11071079）, Natural Science Foundation of Anhui Province （No. 10040606Q47） and Natural Science Foundation of Zhejiang Province （No. Y6110043）. This work of the fourth author is supported by the National Natural Science Foundation of China （No. 10901056）, Science and Technology Commission of Shanghai Municipality （No. 11QA1402200）.
文摘This work is concerned with the nonlinear matrix equation Xs ＋ A＊F（X）A = Q with s ≥ 1. Several sufficient and necessary conditions for the existence and uniqueness of the Hermitian positive semidefinite solution are derived, and perturbation bounds are presented.
基金This work was supported by the Chinese Outstanding Youth Foundation (No. 69925308)Program for Changjiang Scholars and Innovative Research Team in University.
文摘In this note, the matrix equation AV ＋ BW = EVJ is considered, where E, A and B are given matrices of appropriate dimensions, J is an arbitrarily given Jordan matrix, V and W are the matrices to be determined. Firstly, a right factorization of （sE - A）^-1 B is given based on the Leverriver algorithm for descriptor systems. Then based on this factorization and a proposed parametric solution, an alternative parametric solution to this matrix equation is established in terms of the R-controllability matrix of （E, A, B）, the generalized symmetric operator and the observability matrix associated with the Jordan matrix d and a free parameter matrix. The proposed results provide great convenience for many analysis and design problems. Moreover, some equivalent forms are proposed. A numerical example is employed to illustrate the effect of the proposed approach.
文摘An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.
