Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices
The authors discuss the problem of finding error bounds in a perturbed eigenpair \((\widehat\lambda,\widehat x)\) of diagonalizable matrices. An absolute perturbation bound, an extension of \textit{C. Davis} and \textit{W. M. Kahan}'s \(\sin \theta\) theorem for Hermitian matrices [Bull. Am. Math. Soc. 75, 863-868 (1969; Zbl 0175.43204)], and two relative perturbation bounds assuming that \((\widehat\lambda,\widehat x)\) is an exact eigenpair of a perturbed matrix \(D_1AD_2\) not necessarily diagonalizable, where \(D_1\) and \(D_2\) are nonsingular, are derived. A bound on the relative error in \(\widehat\lambda\) and a \(\sin\theta\) theorem based o a relative eigenvalue separation are presented. The perturbation bounds contain both the deviation of \(D_1\) and \(D_2\) from similarity and the deviation of \(D_2\) from identity.
scientific article; zbMATH DE number 1228551