Spectral Perturbation Bounds for Positive Definite Matrices

Let H and H + $\Delta$ H be positive definite matrices. It was shown by Barlow and Demmel and Demmel and Veselic that if one takes a componentwise approach one can prove much stronger bounds on $\lambda_i(H)/\lambda_i(H + \Delta H)$ and the components of the eigenvectors of H and H + $\Delta$ H than by using the standard normwise perturbation theory. Here a unified approach is presented that improves on the results of Barlow, Demmel, and Veselic. It is also shown that the growth factor associated with the error bound on the components of the eigenvectors computed by Jacobi's method grows linearly (rather than exponentially) with the number of Jacobi iterations required for convergence.