The Moore-Penrose and group inverse of square matrices and the Jordan canonical form (Q1373430)

For any \(m\times n\) complex matrix \(A\), \[ (1) \quad AXA=A, \qquad (2) \quad XAX=X, \qquad (3) \quad(AX)^* =AX, \qquad (4) \quad (XA)^* =XA, \] are called Penrose's equations. \textit{R. Penrose} [Proc. Cambridge Phil. Soc. 51, 406-413 (1955; Zbl 0065.24603)] proved that there exists exactly one \(n\times m\) matrix \(X\) which satisfies those equations. \textit{C. Giurescu} and \textit{R. Gabriel} [An. Univ. Timisoara, Ser. Sti. Mat.-Fiz. 2, 103-111 (1964; Zbl 0166.03102)] solved the first two Penrose'e equations for a square matrix, which are transformed into the Jordan canonical form. The main aim of the present paper is to give a general form of a matrix satisfying equations (1), (2) and at least one of Penrose's equations (3) and (4). The other aim is, starting from the Jordan canonical form, a block representation of the group inverse matrix \(X\), i.e. the one satisfying (1), (2) and (5) \(AX= XA\).