The optimal perturbation bounds of the Moore–Penrose inverse under the Frobenius norm

We obtain the optimal perturbation bounds of the Moore–Penrose inverse under the Frobenius norm by using Singular Value Decomposition, which improved the results in the earlier paper [P.-Å. Wedin, Perturbation theory for pseudo-inverses, BIT 13 (1973) 217–232]. In addition, a perturbation bound of the Moore–Penrose inverse under the Frobenius norm in the case of the multiplicative perturbation model is also given.