An Iterative Algorithm for the Reflexive Solution of the General Coupled Matrix Equations

Zhongli Zhou,Guangxin Huang

doi:10.1155/2013/952974

Zhongli Zhou, Guangxin Huang

Open Access

https://doi.org/10.1155/2013/952974

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Journal: TheScientificWorldJournal	Publication Date: Jan 1, 2013
Citations: 11	License type: CC BY 3.0

Affiliation: Chengdu University of Technology

Abstract

The general coupled matrix equations (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 over reflexive matrix solution. When the general coupled matrix equations are consistent over reflexive matrices, the 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 reflexive solution of the general coupled matrix equations can be derived when an appropriate initial matrix is chosen. Furthermore, the unique optimal approximation reflexive solution to a given matrix group in Frobenius norm can be derived by finding the least-norm reflexive solution of the corresponding general coupled matrix equations. A numerical example is given to illustrate the effectiveness of the proposed iterative algorithm.

Highlights

Let P ∈ Rn×n be a generalized reflection matrix; that is, PT = P and P2 = I
The set of all n-by-n reflexive matrices with respect to the generalized reflection matrix P is denoted by Rnr×n(P)
We denote by the superscript T the transpose of a matrix

Summary

Introduction

Wu et al [7, 8] gave the finite iterative solutions to coupled Sylvester-conjugate matrix equations. Huang et al [13] presented a finite iterative algorithms for the one-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions. Yin et al [14] presented a finite iterative algorithms for the twosided and generalized coupled Sylvester matrix equations over reflexive solutions. For any arbitrary initial matrix group, we can obtain a reflexive solution group of Problem 1 within finite iteration steps in the absence of round-off errors. For a special initial matrix group, we can obtain the least Frobenius norm solution of Problem 1. We give the optimal approximate solution group of Problem 2 by finding the least Frobenius norm reflexive solution group of the corresponding general coupled matrix equations.

An Iterative Algorithm for Solving Problem 1

The Solution of Problem 2

Conclusions

The Proof of Lemma 4

The Proof of Lemma 5