A note on unifying absolute and relative perturbation bounds

AbstractPerturbation bounds for invariant subspaces and eigenvalues of complex matrices are presented that lead to absolute as well as a large class of relative bounds. In particular it is shown that absolute bounds (such as those by Davis and Kahan, Bauer and Fike, and Hoffman and Wielandt) and some relative bounds are special cases of `universal' bounds. As a consequence, we obtain a new relative bound for subspaces of normal matrices, which contains a deviation of the matrix from (positive-) definiteness. We also investigate how row scaling affects eigenvalues and their sensitivity to perturbations, and we illustrate how the departure from normality can affect the condition number (with respect to inversion) of the scaled eigenvectors