Revisiting the m-weak core inverse

The $m$-weak core inverse of a complex matrix was introduced by D. E. Ferreyra and Saroj B. Malik in 2024. We have revisited this inverse by using the inverse along two matrices, that is, we have proved that the $m$-weak core inverse of a complex matrix coincides with the inverse along two complex matrices. Moreover, the necessary and sufficient conditions of the $m$-weak core inverse of a complex matrix have been obtained. The one-sided $m$-weak core inverse has been introduced by using the core-EP (EP means Equal Prohection) inverse of $A$.