On the closeness of eigenvalues and singular values for almost normal matrices

Abstract It is proved that a matrix is almost normal if and only if its singular values are close to the absolute values of its eigenvalues. In the special case when the spectral norm and spectral radius are close, it is proved that the dominating eigenvalue is well conditioned. A refinement of a perturbation theorem by Henrici is proved, and its numerical behavior is compared with adaptations of the Gerschgorin theorem. It is specially devised for almost triangular matrices.