Department of Mathematics
Perturbation analysis for antitriangular schur decomposition
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.
Antitriangular Schur form, Condition number, Perturbation analysis
Source Publication Title
SIAM Journal on Matrix Analysis and Applications
Society for Industrial and Applied Mathematics
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.