### Abstract

We study the matrix equation X+ A^{⊤}X^{-1}A=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 language | English |
---|---|

Pages (from-to) | 1187-1192 |

Number of pages | 6 |

Journal | Linear Algebra and Its Applications |

Volume | 435 |

Issue number | 6 |

DOIs | |

State | Published - 15 Sep 2011 |

### Keywords

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

## Fingerprint Dive into the research topics of 'Complex symmetric stabilizing solution of the matrix equation X+ A <sup>⊤</sup>x<sup>-1</sup>A=Q'. Together they form a unique fingerprint.

## Cite this

^{⊤}x

^{-1}A=Q.

*Linear Algebra and Its Applications*,

*435*(6), 1187-1192. https://doi.org/10.1016/j.laa.2011.03.034