Parametrized solutions $X$ of the system $AXA = AY A$ and $A^k Y AX = XAY A^k$

Abstract

Let A and E be n × 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^kEAX = XAEA^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.