Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices

AbstractLet $$\hat \lambda $$ and $$\hat x$$ be a perturbed eigenpair of a diagonalisable matrixA. The problem is to bound the error in $$\hat \lambda $$ and $$\hat \lambda $$ . We present one absolute perturbation bound and two relative perturbation bounds.The absolute perturbation bound is an extension of Davis and Kahan's sin θ Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that $$\hat \lambda $$ and $$\hat x$$ are an exact eigenpair of a perturbed matrixD1AD2, whereD1 andD2 are non-singular, butD1AD2 is not necessarily diagonalisable. We derive a bound on the relative error in $$\hat \lambda $$ and a sin θ theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation ofD1 andD2 from similarity and the deviation ofD2 from identity.