Perturbations of invariant subspaces of unreduced Hessenberg matrices

Non-structured perturbation of invariant subspaces of unreduced, i.e. nonderogatory Hessenberg matrices is considered. Some perturbation results for the generalized eigenvectors and the characteristic polynomial of unreduced upper Hessenberg matrices are given. Two theorems are on the perturbation of invariant subspaces which are somewhat similar to the sinΘsinΘ theorems of Davis and Kahan apart from the residual, which we do not have here. Dense perturbations of unreduced Hessenberg matrices are also considered. Finally, we prove an invariant subspace perturbation theorem for nonderogatory matrices.