Matrix equations, Index, G-Drazin inverses, Generalized inverses
Let $A$ and $E$ be $n \times n$ given complex matrices. This paper provides a necessary and sufficient condition for the solvability to the matrix equation system given by $AXA=AEA$ and $A^k E A X = X A E A^k$, for $k$ being the index of $A$. In addition, its general solution is derived in terms of a G-Drazin inverse of $A$. As consequences, new representations are obtained for the set of all G-Drazin inverses; some interesting applications are also derived to show the importance of the obtained formulas.
Ferreyra, David; Lattanzi, Marina; Levis, Fabián; and Thome, Néstor.
"Parametrized solutions $X$ of the system $AXA = AY A$ and $A^k Y AX = XAY A^k$",
Electronic Journal of Linear Algebra,
Volume 35, pp. 503-510.