Norm estimations for the Moore–Penrose inverse of multiplicative perturbations of matrices

Let B   be a multiplicative perturbation of A∈Cm×nA∈Cm×n given by B=D1⁎AD2, where D1∈Cm×mD1∈Cm×m and D2∈Cn×nD2∈Cn×n are both nonsingular. New norm upper bounds for ‖B†−A†‖F‖B†−A†‖F and ‖B†−A†‖2‖B†−A†‖2 are derived, where A†A† and B†B† are the Moore–Penrose inverses of A and B, respectively. The main results of Meng and Zheng (2015) [10] are improved in the cases of the Frobenius norm and the spectral norm.