On the condition of a complex eigenvalue under real perturbations
The paper focuses on the condition number for a complex eigenvalue of a real matrix under real perturbations, showing that this number is never smaller than \(1/\sqrt{2}\) times the corresponding condition number with respect to complex perturbations. The first part represents an introduction concerning the errors, the backward stability of an algorithm, the backward error, notions which are used in the sequel. The second part concerns the real and complex eigenvalue condition numbers. The authors present and prove a theorem which characterizes explicitly the absolute condition number \({\mathcal C}^R(\lambda)\) for \(\lambda\in \mathbb{C}\) -- a simple eigenvalue of \(A\in\mathbb{R}^{n\times n}\), and provide a bound on how much \({\mathcal C}^C(\lambda)\) can exceed \({\mathcal C}^R(\lambda)\). The third part deals with real-part and imaginary-part condition numbers, showing that at least one of them must be within a factor \(1/\sqrt{2}\) of the complex perturbation condition number. The fourth section contains some generalizations of the above results to real perturbations of complex matrices. The last section presents the conclusions concerning the ill-conditioning of the real part or the imaginary part of an eigenvalue of a real matrix under real perturbations.
scientific article; zbMATH DE number 2153698