Representations of generalized inverses and Drazin inverse of partitioned matrix with Banachiewicz-Schur forms (Q1793766)

Summary: Representations of \(\left\{1,2, 3\right\}\)-inverses, \(\left\{1,2, 4\right\}\)-inverses, and Drazin inverse of a partitioned matrix \(M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}\) related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which \(\left\{1,2, 3\right\}\)-inverses, \(\left\{1,2, 4\right\}\)-inverses, and group inverse of a \(2 \times 2\) block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of \textit{D. S. Cvetković-Ilić} [Appl. Math. Comput. 213, No. 1, 18--24 (2009; Zbl 1170.15002)] are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.