ACTA issues

Unified approach to the reverse order rule for generalized inverses

Dragan S. Djordjević

Acta Sci. Math. (Szeged) 67:3-4(2001), 761-776
2816/2009

 Abstract. In this paper we consider the reverse order rule of the form $(AB)_{K,L}^{(2)} =B_{T,S}^{(2)} A_{M,N}^{(2)}$ for outer generalized inverses with prescribed range and kernel. As corollaries, we get generalizations of the well-known results of Bouldin (SIAM J. Appl. Math. {\bf25} (1973), 489--495; Recent Applications of Generalized Inverses'', Pitman Ser. Res. Notes in Math. vol {\bf66}, (1982), 233--248) and Izumino (Tohoku Math. J. {\bf34} (1982), 43--52) for the ordinary Moore--Penrose inverse, and Sun and Wei (SIAM J. Matrix Anal. Appl. {\bf19} (1998), 772--775) for the weighted Moore--Penrose inverse. Results of Bouldin (the second paper mentioned above) for the reverse order rule for the Drazin inverse are improved. Finally, necessary and suficient conditions such that the reverse order rule holds for the group inverse are introduced. AMS Subject Classification (1991): 47A05, 15A09 Received June 20, 2000, and in revised form January 15, 2001. (Registered under 2816/2009.)