  It is well known that a square complex matrix is called EP if it commutes
with its Moore-Penrose inverse. In this paper, new classes of matrices which
extend this concept are characterized. For that, we consider commutative
equalities given by matrices of arbitrary index and generalized inverses
recently investigated in the literature. More specifically, these classes are
characterized by expressions of type $A^mX=XA^m$, where $X$ is an outer inverse
of a given complex square matrix $A$ and $m$ is an arbitrary positive integer.
The relationships between the different classes of matrices are also analyzed.
Finally, a picture presents an overview of the overall studied classes.
