An iterative algorithm for the generalized reflexive solutions of the general coupled matrix equations

Feng Yin,Ke Guo,Guang-Xin Huang

doi:10.1186/1029-242x-2013-280

Abstract

The general coupled matrix equations ∑ j = 1 q A i j X j B i j = M i ,i=1,2,…,p (including the generalized coupled Sylvester matrix equations as special cases) have numerous applications in control and system theory. In this paper, an iterative algorithm is constructed to solve the general coupled matrix equations and their optimal approximation problem over generalized reflexive matrix solution ( X 1 , X 2 ,…, X q ). When the general coupled matrix equations are consistent over generalized reflexive matrices, the generalized reflexive solution can be determined automatically by the iterative algorithm within finite iterative steps in the absence of round-off errors. The least Frobenius norm generalized reflexive solution of the general coupled matrix equations can be derived when an appropriate initial matrix group is chosen. Furthermore, the unique optimal approximation generalized reflexive solution ( X ˆ 1 , X ˆ 2 ,…, X ˆ q ) to a given matrix group ( X 1 0 , X 2 0 ,…, X q 0 ) in Frobenius norm can be derived by finding the least-norm generalized reflexive solution ( X ˜ 1 ∗ , X ˜ 2 ∗ ,…, X ˜ q ∗ ) of the corresponding general coupled matrix equations ∑ j = 1 q A i j X ˜ j B i j = M ˜ i , i=1,2,…,p, where X ˜ j = X j − X j 0 , M ˜ i = M i − ∑ j = 1 q A i j X j 0 B i j . A numerical example is given to illustrate the effectiveness of the proposed iterative algorithm.MSC:15A18, 15A57, 65F15, 65F20.

Highlights

For given matrices Aij ∈ Rri×mj , Bij ∈ Rnj×si , and Mi ∈ Rri×si , find a generalized reflexive matrix solution group
Let P ∈ Rm×m and Q ∈ Rn×n be two real generalized reflection matrices, i.e., PT = P, P = Im, QT = Q, Q = In, In denotes the n order identity matrix
Novel gradient-based iterative (GI) method [ – ] and least-squares-based iterative methods [, ] with highly computational efficiencies for solving matrix equations are presented and have good stability performances based on the hierarchical identification principle, which regards the unknown matrix as the system parameter matrix to be identified

Summary

Introduction

For given matrices Aij ∈ Rri×mj , Bij ∈ Rnj×si , and Mi ∈ Rri×si , find a generalized reflexive matrix solution group Yin et al [ ] presented a finite iterative algorithm for the two-sided and generalized coupled Sylvester matrix equations over reflexive solutions. For an arbitrary initial matrix group, we can obtain a generalized reflexive solution group of Problem I

Results

Conclusion