Generalized core inverses of matrices

In this paper, we introduce two new generalized inverses of matrices, namely, the $bra{i}{m}$-core inverse and the $pare{j}{m}$-core inverse. The $bra{i}{m}$-core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler cite{BT} and the core-EP inverse defined by Manjunatha Prasad and Mohana cite{MM}. The $pare{j}{m}$-core inverse of a complex matrix extends the notions of the core inverse and the ${rm DMP}$-inverse defined by Malik and Thome cite{MT}. Moreover, the formulae and properties of these two new concepts are investigated by using matrix decompositions and matrix powers.