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.