On the condition of a complex eigenvalue under real perturbations

We investigate the condition number for a complex eigenvalue of a real matrix under real perturbations. Based on an explicit formula, it is shown that this number is never smaller than 1/root2 times the corresponding condition number with respect to complex perturbations. This result can be generalized to the condition number of an arbitrary complex-valued function under real perturbations. This extends to related condition numbers.