Perturbation Analysis for Antitriangular Schur Decomposition

Let Z be an n× n complex matrix. A decomposition Z=ŪM UH 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(MXL-X̄U M), Aup(MTXL -X̄U MT)) \|F, where XL and XU are the strictly lower triangular and upper triangular parts of X, XN=XL+XU, 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.