Weak CMP inverses

We consider extended systems of matrix equations than well-known systems for presenting the CMP inverse in terms of a minimal rank weak Drazin inverse and minimal rank right weak Drazin inverse. Solutions of new extended systems present new kinds of generalized inverses and they are called a weak CMP inverse and a right weak CMP inverse. Beside that the CMP, MPCEP and *CEPMP inverses are special types of weak CMP and right weak CMP inverses, we study two particular classes of weak CMP inverse which are new in literature. A number of characterizations and representations for weak CMP inverse are developed. Perturbation formulae and perturbation bounds for weak CMP inverse are proved. Applications of the weak CMP inverse for solving linear equations are presented. Consequently, we obtain the classical result about application of the Moore–Penrose inverse in solving linear equation.