Complex symmetric stabilizing solution of the matrix equation X+ A x-1A=Q

Chun Hua Guo, Yueh Cheng Kuo, Wen-Wei Lin*

*Corresponding author for this work

Research output: Contribution to journalArticle

15 Scopus citations

Abstract

We study the matrix equation X+ AX-1A=Q, where A is a complex square matrix and Q is complex symmetric. Special cases of this equation appear in Green's function calculation in nano research and also in the vibration analysis of fast trains. In those applications, the existence of a unique complex symmetric stabilizing solution has been proved using advanced results on linear operators. The stabilizing solution is the solution of practical interest. In this paper we provide an elementary proof of the existence for the general matrix equation, under an assumption that is satisfied for the two special applications. Moreover, our new approach here reveals that the unique complex symmetric stabilizing solution has a positive definite imaginary part. The unique stabilizing solution can be computed efficiently by the doubling algorithm.

Original languageEnglish
Pages (from-to)1187-1192
Number of pages6
JournalLinear Algebra and Its Applications
Volume435
Issue number6
DOIs
StatePublished - 15 Sep 2011

Keywords

  • Complex symmetric solution
  • Doubling algorithm
  • Nonlinear matrix equation
  • Stabilizing solution

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