Explicit Solutions of the Matrix Equation $\sum {A^i XD_i } = C$

1992 ◽  
Vol 13 (4) ◽  
pp. 1123-1130 ◽  
Author(s):  
Harald K. Wimmer
Keyword(s):  
Matrix Equation ◽  
Explicit Solutions ◽  
The Matrix

scholarly journals Re-nnd SOLUTIONS OF THE MATRIX EQUATION AXB=C

2008 ◽  
Vol 84 (1) ◽  
pp. 63-72 ◽  
Author(s):  
DRAGANA S. CVETKOVIĆ-ILIĆ
Keyword(s):  
Inverse Problem ◽  
Linear Algebra ◽  
Matrix Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Explicit Solutions ◽  
Matrix Inverse ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Special Case

AbstractIn this article we consider Re-nnd solutions of the equation AXB=C with respect to X, where A,B,C are given matrices. We give necessary and sufficient conditions for the existence of Re-nnd solutions and present a general form of such solutions. As a special case when A=I we obtain the results from a paper of Groß (‘Explicit solutions to the matrix inverse problem AX=B’, Linear Algebra Appl.289 (1999), 131–134).

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

scholarly journals Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA∗=B

2009 ◽  
Vol 431 (12) ◽  
pp. 2359-2372 ◽  
Author(s):  
Yonghui Liu ◽  
Yongge Tian ◽  
Yoshio Takane
Keyword(s):  
Matrix Equation ◽  
The Matrix
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