Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

Perturbation analysis for antitriangular schur decomposition

Language

English

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 ∥XN∥ F=1 ∥(Aup(MX L - X̄ UM), Aup(M TX L - X̄ UM 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. © 2012 Society for Industrial and Applied Mathematics.

Keywords

Antitriangular Schur form, Condition number, Perturbation analysis

Publication Date

2012

Source Publication Title

SIAM Journal on Matrix Analysis and Applications

Volume

33

Issue

2

Start Page

325

End Page

335

Publisher

Society for Industrial and Applied Mathematics

DOI

10.1137/110841370

Link to Publisher's Edition

http://dx.doi.org/10.1137/110841370

ISSN (print)

08954798

ISSN (electronic)

10957162

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