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
ISSN (print)
08954798
ISSN (electronic)
10957162
Recommended Citation
Chen, Xiao Shan, Wen Li, and Michael K. Ng. "Perturbation analysis for antitriangular schur decomposition." SIAM Journal on Matrix Analysis and Applications 33.2 (2012): 325-335.