Strong DMP inverse

A new-type matrix inverse based on the Hartwig-Spindelböck decomposition was investigated, which is related to the DMP inverse, is the abbreviation of the Drazin Moore-Penrose inverse, here we call this generalized inverse as the strong DMP inverse. We established the relationships between this new-type inverse and other matrix generalized inverses. We proved that the strong DMP inverse of a square complex matrix is a special type of outer inverse, where the restricted column space is $\mathcal{R}(A^k)$ and restricted null space is $\mathcal{N}(A{A}^D{\alpha }_A)$, where ${\alpha }_A=A{A}^{†}+(A^{†}{)}^{\ast }(E_n-A{A}^{†})$. The one-sided strong DMP inverses were introduced from the perspectives of the column space, null space, and index of a matrix. The general expressions of the left (right) strong DMP inverse of $A$ were given. We answered the question of when the left strong DMP inverse is consistent with the right strong DMP inverse. Based on the relationship between the left strong DMP inverse and the right DMP inverse, the necessary and sufficient conditions for the existence of the strong DMP inverse were given.