Expression of the Drazin and MP-inverse of partitioned matrix and quotient identity of generalized Schur complement

In this paper we give representations of the Drazin and MP-inverse of a 2×2 block matrix and quotient identities for the generalized Schur complement of a partitioned 3×3 matrix under conditions different than those used in recent papers on the subject. We present numerical examples to illustrate our results.     

A comment on some recent results concerning the reverse order law for {1, 3, 4}-inverses

This paper has been motivated by the one of Liu and Yang [D. Liu, H. Yang, The reverse order law for {1, 3, 4}-inverse of the product of two matrices, Appl. Math. Comp. 215 (12) (2010) 4293-4303] in which the authors consider separately the cases when (AB){1,3,4}⊆B{1,3,4}·A{1,3,4} and (AB){1,3,4}=B{1,3,4}·A{1,3,4}, where A∈C^n×m and B∈C^m×n. Here we prove that (AB){1,3,4}⊆B{1,3,4}·A{1,3,4} is actually equivalent to (AB){1,3,4}=B{1,3,4}·A{1,3,4}. We show that (AB){1,3,4}⊆B{1,3,4}·A{1,3,4} can only be possible if and in this case, we present purely algebraic necessary and sufficient conditions for this inclusion to hold. Also we give some new characterizations of B{1,3,4}·A{1,3,4}⊆(AB){1,3,4}.     

Improvements on the reverse order laws

In many papers concerning properties of generalized inverses in different settings, we can find the results with many redundant instances of assuming regularity of certain elements. We have made an effort to widen the range of applicability of concrete results by considering more general cases of the problems without imposing any such additional assumptions. This was the main motivation for this paper, and we present several significant improvements of the reverse order laws on $\{1,3\}$ and $\{1,2,3\}$-inverses in the ring setting.