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A multiplicative perturbation  M  of a matrix  T  has the form, where  E  and  F  are square matrices. It is proved that every acute perturbation is essentially a strong perturbation, which is a type of multiplicative perturbation. It is also proved that for every multiplicative perturbationis a strong perturbation if and only if it is a weak perturbation and is rank-preserving. Some norm equations for the Moore-Penrose inverse are derived in the framework of the weak perturbation, through which some norm upper bounds forare obtained. As an application, the perturbation estimation for the solution to the least squares problems is provided. The sharpness of the newly obtained upper bounds are illustrated by several numerical examples.

Norm estimations for the Moore-Penrose inverse of the weak perturbation of matrices