## Abstract

Let Z be an n × n complex matrix. A decomposition Z = ŪMU ^{H} is called an antitriangular Schur decomposition of Z if U is an n × n unitary matrix and M is an n × n antitriangular matrix. The antitriangular Schur decomposition is a useful tool for solving palindromic eigenvalue problems. However, there is no perturbation result for an antitriangular Schur decomposition in the literature. The main contribution of this paper is to give a perturbation bound of such decomposition and show that the bound depends inversely on f(M) := min _{∥X}N∥ _{F=1} ∥(Aup(MX _{L} - X̄ _{U}M), Aup(M ^{T}X _{L} - X̄ _{U}M ^{T}))∥ _{F}, where X _{L} and X _{U} are the strictly lower triangular and upper triangular parts of X, X _{N} = X _{L} + X _{U}, and Aup(Y ) denotes the strictly upper antitriangular part of Y. The quantity √2/f(M) can be used to characterize the condition number of the decomposition, i.e., when √2/f(M) is large (or small), the decomposition problem is ill-conditioned (or well-conditioned). Numerical examples are presented to illustrate the theoretical result.

Original language | English |
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Pages (from-to) | 325-335 |

Number of pages | 11 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 33 |

Issue number | 2 |

DOIs | |

Publication status | Published - 17 Apr 2012 |

## Scopus Subject Areas

- Analysis

## User-Defined Keywords

- Antitriangular Schur form
- Condition number
- Perturbation analysis