scholarly journals A New Solution to the Matrix EquationX−AX¯B=C

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Caiqin Song
Keyword(s):  
Unique Solution ◽  
Canonical Form ◽  
Matrix Equation ◽  
Explicit Solution ◽  
Symmetric Operator ◽  
Operator Matrix ◽  
Numerical Example ◽  
Controllability Matrix ◽  
The Matrix

We investigate the matrix equationX−AX¯B=C. For convenience, the matrix equationX−AX¯B=Cis named as Kalman-Yakubovich-conjugate matrix equation. The explicit solution is constructed when the above matrix equation has unique solution. And this solution is stated as a polynomial of coefficient matrices of the matrix equation. Moreover, the explicit solution is also expressed by the symmetric operator matrix, controllability matrix, and observability matrix. The proposed approach does not require the coefficient matrices to be in arbitrary canonical form. At the end of this paper, the numerical example is shown to illustrate the effectiveness of the proposed method.

scholarly journals Polynomial Solutions to the Matrix EquationX−AXTB=C

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Caiqin Song ◽  
Jun-e Feng
Keyword(s):  
Control Theory ◽  
Unique Solution ◽  
Canonical Form ◽  
Matrix Equation ◽  
Explicit Solution ◽  
Symmetric Operator ◽  
Operator Matrix ◽  
Polynomial Solutions ◽  
Controllability Matrix ◽  
The Matrix

Solutions are constructed for the Kalman-Yakubovich-transpose equationX−AXTB=C. The solutions are stated as a polynomial of parameters of the matrix equation. One of the polynomial solutions is expressed by the symmetric operator matrix, controllability matrix, and observability matrix. Moreover, the explicit solution is proposed when the Kalman-Yakubovich-transpose matrix equation has a unique solution. The provided approach does not require the coefficient matrices to be in canonical form. In addition, the numerical example is given to illustrate the effectiveness of the derived method. Some applications in control theory are discussed at the end of this paper.

The explicit solution of the matrix equation AX−XB=C

1995 ◽  
Vol 16 (12) ◽  
pp. 1133-1141 ◽  
Author(s):  
Chen Yuming ◽  
Xiao Heng
Keyword(s):  
Matrix Equation ◽  
Explicit Solution ◽  
The Matrix

An explicit solution to the matrix equation AV+BW=EV J

2007 ◽  
Vol 5 (1) ◽  
pp. 47-52 ◽  
Author(s):  
Aiguo Wu ◽  
Guangren Duan ◽  
Bin Zhou
Keyword(s):  
Matrix Equation ◽  
Explicit Solution ◽  
The Matrix

Further Solutions of a Yang-Baxter-like Matrix Equation

2013 ◽  
Vol 3 (4) ◽  
pp. 352-362 ◽  
Author(s):  
Jiu Ding ◽  
Chenhua Zhang ◽  
Noah H. Rhee
Keyword(s):  
Canonical Form ◽  
Infinite Number ◽  
Matrix Equation ◽  
Number Of Solutions ◽  
Jordan Canonical Form ◽  
The Matrix ◽  
Square Matrix

AbstractThe Yang-Baxter-like matrix equation AXA = XAX is reconsidered, and an infinite number of solutions that commute with any given complex square matrix A are found. Our results here are based on the fact that the matrix A can be replaced with its Jordan canonical form. We also discuss the explicit structure of the solutions obtained.

scholarly journals Perturbation Analysis for the Matrix EquationX−∑i=1mAi∗XAi+∑j=1nBj∗XBj=I

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Xue-Feng Duan ◽  
Qing-Wen Wang
Keyword(s):  
Perturbation Analysis ◽  
Matrix Equation ◽  
Positive Definite ◽  
Perturbation Bound ◽  
Numerical Example ◽  
Positive Definite Solution ◽  
The Matrix ◽  
Definite Solution

We consider the perturbation analysis of the matrix equationX−∑i=1mAi∗XAi+∑j=1nBj∗XBj=I. Based on the matrix differentiation, we first give a precise perturbation bound for the positive definite solution. A numerical example is presented to illustrate the sharpness of the perturbation bound.

scholarly journals Explicit solution of the matrix equation AXB − CXD = E

1989 ◽  
Vol 121 ◽  
pp. 333-344 ◽  
Author(s):  
Vicente Hernández ◽  
Maite Gassó
Keyword(s):  
Matrix Equation ◽  
Explicit Solution ◽  
The Matrix

scholarly journals Convergence analysis of gradient-based iterative algorithms for a class of rectangular Sylvester matrix equations based on Banach contraction principle

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Adisorn Kittisopaporn ◽  
Pattrawut Chansangiam ◽  
Wicharn Lewkeeratiyutkul
Keyword(s):  
Spectral Radius ◽  
Matrix Equation ◽  
Iterative Algorithms ◽  
Banach Contraction Principle ◽  
Contraction Principle ◽  
Convergence Factor ◽  
Sylvester Matrix ◽  
Iteration Matrix ◽  
Banach Contraction ◽  
The Matrix

AbstractWe derive an iterative procedure for solving a generalized Sylvester matrix equation $AXB+CXD = E$ A X B + C X D = E , where $A,B,C,D,E$ A , B , C , D , E are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation $AXB=C$ A X B = C , the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.

Solution of the matrix equation AX + XB = C [F4]

1972 ◽  
Vol 15 (9) ◽  
pp. 820-826 ◽  
Author(s):  
R. H. Bartels ◽  
G. W. Stewart
Keyword(s):  
Matrix Equation ◽  
The Matrix
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