On the Kahan-Parlett-Jiang theorem -- a globally optimal backward perturbation error for two-sided invariant subspaces (Q2185840)

Let \(A\in \mathbb{C}^{n\times n}\) and two subspaces \(\mathbf{X}\) and \(\mathbf{Y}\) of dimension \(m\) be given, and let \(X_m\) and \(Y_m\) be orthonormal basis matrices for \(\mathbf{X}\) and \(\mathbf{Y}\) respectively. In the field of large non-Hermitian eigenvalue problems the following task investigated by \textit{W. Kahan} et al. [SIAM J. Numer. Anal. 19, 470--484 (1982; Zbl 0483.65024)] is very important: given matrices \(M_m\) and \(L_m=(Y_m^{H}X_m)^{-1}M_m^{H}(Y_m^{H}X_m)\), one seeks matrices \(E\in \mathbb{C}^{n\times n}\) of minimal norm, such that \[ \begin{array}{l} (A+E)X_m=X_mL_m; \\ (A+E)^{H}Y_m=Y_mM_m. \end{array} \] Evidently, the matrix \(E\) is not globally optimal in general. In the present article the authors revisit this problem and derive a globally optimal backward perturbation error for given two-sided approximate invariant subspaces with respect to the Frobenius norm related to solving the optimization problem: \[ \min_{L_m\in \mathbb{C}^{m\times m},\, M_m = (X_m^H Y_m)^{-1} L_m^H (X_m^H Y_m)} \|E\|_F. \] Theorem 2.4 enhances the Kahan-Parlett-Jiang theorem. An algorithm for calculating the F-norm of the globally optimal perturbation matrix is also given. Numerical experiments demonstrate the effectiveness of the theoretical results.