  The purpose of this paper is to extend the definition of the m-weak group
inverse from a square matrix to a rectangular matrix, called the W-weighted
m-weak group inverse. This new generalized inverse is also a generalization of
the weak group inverse, generalized group inverse, Drazin inverse, weighted
weak group inverse and W-weighted Drazin inverse. Furthermore, we discuss some
properties, characterizations and representations of the W-weighted m-weak
group inverse, as well as its applications in solving the matrix equation. Some
of the results available in the paper not only recover a few results of the
m-weak group inverse, weighted weak group, etc., but also some of them are
novel for these generalized inverses.
