The {1,2,3,1m}-inverses: A generalization of core inverses for matrices

•In this paper, for any complex square matrix A (with index m), we introduce the new concept of {1,2,3,1m}-inverses as a generalization of the core inverse. The set of all {1,2,3,1m}-inverses of a matrix is described.•It is proved that A has a unique {1,2,3,1m}-inverse for some m if and only if it has index 0 or 1, in which case the core inverse is exactly its unique {1,2,3,1m}-inverse.•EP properties of matrices are characterized by using {1,2,3,1m}-inverses. This paper concerns a new generalized inverse for matrices of an arbitrary index. It is proved that every complex square matrix A possesses a {1,2,3}-inverse X such that XAm+1=Am for some integer m. We shall call such X a {1,2,3,1m}-inverse of A. A notable result is that A has a unique {1,2,3,1m}-inverse if and only if it has index 0 or 1, in which case ▪ is exactly its unique {1,2,3,1m}-inverse. For a matrix with an arbitrary index, the set of all its {1,2,3,1m}-inverses is completely determined. Some new characterizations of EP matrices, generalized EP matrices and m-EP matrices are established by using their {1,2,3,1m}-inverses.