scholarly journals Finite iterative algorithms for the generalized reflexive and anti-reflexive solutions of the linear matrix equation AXB = C

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 2151-2162 ◽  
Author(s):  
Xiang Wang ◽  
Xiao-Bin Tang ◽  
Xin-Geng Gao ◽  
Wu-Hua Wu
Keyword(s):  
Iterative Method ◽  
Matrix Equation ◽  
Iterative Algorithms ◽  
Iterative Solution ◽  
Necessary Condition ◽  
Linear Matrix ◽  
Linear Matrix Equation ◽  
Reflexive Matrix ◽  
Roundoff Errors ◽  
Sufficient And Necessary Condition

In this paper, an iterative method is presented to solve the linear matrix equation AXB = C over the generalized reflexive (or anti-reflexive) matrix X (A ? Rpxn, B ? Rmxq, C ? Rpxq, X ? Rnxm). By the iterative method, the solvability of the equation AXB = C over the generalized reflexive (or anti-reflexive) matrix can be determined automatically. When the equation AXB = C is consistent over the generalized reflexive (or anti-reflexive) matrix X, for any generalized reflexive (or anti-reflexive) initial iterative matrix X1, the generalized reflexive (anti-reflexive) solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized reflexive (or anti-reflexive) iterative solution of AXB = C can be derived when an appropriate initial iterative matrix is chosen. A sufficient and necessary condition for whether the equation AXB = C is inconsistent is given. Furthermore, the optimal approximate solution of AXB = C for a given matrix X0 can be derived by finding the least-norm generalized reflexive (or anti-reflexive) solution of a new corresponding matrix equation AXB = C. Finally, several numerical examples are given to support the theoretical results of this paper.

scholarly journals An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix EquationsAXB=E,CXD=F

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Deqin Chen ◽  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Approximate Solution ◽  
Iterative Algorithm ◽  
Matrix Equation ◽  
Iterative Solution ◽  
Linear Matrix ◽  
Optimal Approximate Solution ◽  
Linear Matrix Equation ◽  
Numerical Examples ◽  
Reflexive Matrix ◽  
The Matrix

An iterative algorithm is constructed to solve the linear matrix equation pairAXB=E, CXD=Fover generalized reflexive matrixX. When the matrix equation pairAXB=E, CXD=Fis consistent over generalized reflexive matrixX, for any generalized reflexive initial iterative matrixX1, the generalized reflexive solution can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors. The unique least-norm generalized reflexive iterative solution of the matrix equation pair can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate solution ofAXB=E, CXD=Ffor a given generalized reflexive matrixX0can be derived by finding the least-norm generalized reflexive solution of a new corresponding matrix equation pairAX̃B=Ẽ, CX̃D=F̃withẼ=E-AX0B, F̃=F-CX0D. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

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