Perturbation Analysis for Antitriangular Schur Decomposition

Let Z be an $n\times n$ complex matrix. A decomposition $Z=\overline{U}M U^H$ is called an antitriangular Schur decomposition of Z if U is an $n\times n$ unitary matrix and M is an $n\times n$ 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 $f(M):= \min_{\| X_N \|_F = 1} \| (\mbox{Aup}(MX_L-\overline{X}_U M), \mbox{Aup}(M^TX_L -\overline{X}_U M^T)) \|_F$, where $X_L$ and $X_U$ are the strictly lower triangular and upper triangular parts of $X, X_N=X_L+X_U,$ and $\mbox{Aup}(Y)$ denotes the strictly upper antitriangular part of Y. The quantity $\sqrt{2}/f(M)$ can be used to characterize the condition number of the decomposition, i.e., when $\sqrt{2}/f(M)$ is large (or small), the decomposition proble...