The projection m-weak group inverse

This paper introduces the projection $m$-weak group inverse (or P-$m$-WG inverse) for square complex matrices as a linear combination of the $m$-WG inverse $A^{weak_m}$ and its associated projectors $A^{weak_m}A$ and $AA^{weak_m}$. We establish the uniqueness of the P-$m$-WG inverse for any matrix with an arbitrary index and provide several characterizations and properties of this generalized inverse. We explore the connections between the P-$m$-WG inverse of a matrix and it´s other known generalized inverses by using the core-EP decomposition of the matrix.