A generalization of the m -weak core inverse

In order to unify several notions of the core inverse and its extensions, including the DMP inverse, the core–EP inverse, the weak core inverse, the [Formula: see text]-weak core inverse ([Formula: see text]-WCI), the [Formula: see text]-weak group MP inverse, and the dual weak MPCEP inverse, we introduced the weak [Formula: see text]-WCI ([Formula: see text]-[Formula: see text]-WCI) in terms of the minimal rank weak Drazin inverse of [Formula: see text] and the orthogonal projector [Formula: see text]. The [Formula: see text]-[Formula: see text]-WCI is characterized as a solution to several systems of matrix equations. We derive canonical forms and representations of the [Formula: see text]-[Formula: see text]-WCI. Applications to solving linear matrix equations are presented, together with explicit solution formulas. Numerical examples illustrate the theoretical results.