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 AkEAX = XAEAk, 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.

GDMP-inverses of a matrix and their duals

“…Before that, we state some necessary properties. (p2) ∆ k−1 is a G-Drazin inverse of (ΣK) k−1 , (see [26,Lemma 3.1]).…”

“…ii) =⇒ (iii) From[26, Theorem 3.2], Z ∈ A{GD} if and only if for arbitrary T, W ,Z = A GD + (I − P A k )T (I − P A ) + (I − Q A )W (I − P A k ), where P A k = A k (A GD ) k , P A = AA GDand Q A = A GD A. Now, it is easy to see that, for arbitrary T and W , X satisfies equations of system (4.1) becauseP A A k = A k and P A k A k = A k .…”

From projectors to 1MP and MP1 generalized inverses and their induced partial orders

“…then the Drazin inverse of A is called the group inverse of A and is denoted by A # . A detailed analysis of all these generalized inverses can be found, for example, in [1,3,4,12,13].…”

Solving an Open Problem About the G-Drazin Partial Order

“…However, this necessary condition could not be stated as a characterization and the validity (or not) of the converse implication was posed as an open problem. By using recent results proved by the authors in [9], we are in position to solve completely this problem. This paper is organized as follows.…”