editorial linear and nonlinear matrix equationsabstract and applied analysis =1 2 + =1 2 is minimum....

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 741430, 2 pageshttp://dx.doi.org/10.1155/2013/741430

EditorialLinear and Nonlinear Matrix Equations

Masoud Hajarian,1 Qing-Wen Wang,2 and Vejdi I. Hasanov3

1 Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, General Campus,Evin, Tehran 19839, Iran

2Department of Mathematics, Shanghai University, 99 Shangda Road, Shanghai 200444, China3 Faculty of Mathematics and Informatics, Konstantin Preslavsky University of Shumen, 9700 Shumen, Bulgaria

Correspondence should be addressed to Masoud Hajarian; m [email protected]

Received 23 December 2013; Accepted 23 December 2013

Copyright © 2013 Masoud Hajarian et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Matrix equations have practical applications in many areassuch as computational mathematics, biology, electricity,dynamic programming, stochastic filtering, statistics, solidmechanics, and control and system theory. In recent years,a large number of papers have studied several linear andnonlinear matrix equations.

This special issue is devoted to publishing the latest andsignificant results on linear and nonlinear matrix equationson all their aspects. Its goals are to highlight recent advancesand developments on themany facets, techniques, and resultsof linear and nonlinear matrix equations.

The topics included in this special issue are iterativesolutions of matrix equations, closed-form solutions andsolvability of matrix equations, quaternion matrix equations,and perturbation analysis of matrix equations.

We received nineteen papers in the interdisciplinaryresearch fields.This special issue includes seven high quality-peer-reviewed articles.

In the following, we briefly review each of the papers thatare published.

In the paper entitled “Norm-constrained least-squaressolutions to the matrix equation 𝐴𝑋𝐵 = 𝐶,” A. Xu and B.Z. Peng propose an iterative method to compute the least-squares solutions of the matrix equation 𝐴𝑋𝐵 = 𝐶 over thenorm inequality constraint.

In the paper entitled “Iterative solution to a system ofmatrix equations,” Y. Lin and Q. W. Wang introduce anefficient iterative algorithm to solve the system of linearmatrix equations 𝐴

1𝑋1𝐵1+ 𝐴2𝑋2𝐵2= 𝐸 and 𝐶

1𝑋1𝐷1+

𝐶2𝑋2𝐷2= 𝐹, with real matrices 𝑋

1and 𝑋

2. By this iterative

algorithm, the solvability of the system can be determinedautomatically.

In the paper entitled “existence and uniqueness of thepositive definite solution for the matrix equation𝑋 = 𝑄 +𝐴∗(𝑋 − 𝐶)

−1𝐴,” D. Gao proves that the matrix equation 𝑋 =

𝑄+𝐴∗(𝑋 −𝐶)

−1𝐴 has a unique positive definite solution via

variable replacement and fixed point theorem. Also the basicfixed point iteration for the matrix equation is given.

In the paper entitled “Perturbation analysis of the nonlin-ear matrix equation 𝑋 − ∑𝑚

𝑖=1𝐴∗

𝑖𝑋𝑝𝑖𝐴𝑖= 𝑄,” J. Li derives

two perturbation bounds and the backward error of anapproximate solution to the nonlinear matrix equation 𝑋 −∑𝑚

𝑖=1𝐴∗

𝑖𝑋𝑝𝑖𝐴𝑖= 𝑄 with 0 < 𝑝

𝑖< 1. Also explicit expressions

of the condition number for thematrix equation are obtained.In the paper entitled “A Note on the Τ-Stein Matrix

Equation,” C. Y. Chiang proposes a novel approach to thenecessary and sufficient conditions for the unique solvabilityof the solution 𝑋 of the Τ-Stein matrix equation for squarecoefficient matrices in terms of the analysis of the spectra𝜎(𝐴Τ𝐵).

In the paper entitled “On the low-rank approximationarising in the generalized karhunen-loeve transform,” X. F.Duan et al. consider the low-rank approximation problemarising in the generalized Karhunen-Loeve transform. Asufficient condition for the existence of a solution is derived,and the analytical expression of the solution is given.

In the paper entitled “The hermitian 𝑅-conjugate gen-eralized Procrustes problem,” H. X. Chang et al. considerthe Hermitian 𝑅-conjugate generalized Procrustes prob-lem to find Hermitian 𝑅-conjugate matrix 𝑋 such that

2 Abstract and Applied Analysis

∑𝑝

𝑘=1‖𝐴𝑘𝑋 − 𝐶

𝑘‖2+ ∑𝑞

𝑙=1‖𝑋𝐵𝑙− 𝐷𝑙‖2 is minimum. The

expression of the solution to Hermitian 𝑅-conjugate gener-alized Procrustes problem is derived.

Acknowledgments

The editors of this special issue would like to express theirgratitude to the authors who have submitted manuscripts forconsideration. They also thank the many individuals whoserved as referees of the submitted manuscripts.

Masoud HajarianQing-Wen WangVejdi I. Hasanov

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