Group invertible block matrices

Abstract

Let M=\left(\begin{array}{ll}A& B\\ C& D\end{array}\right) (A and D are square) be a 2 × 2 block matrix over a skew field, where A is group invertible. Let S=D-CA^#B denote the generalized Schur complement of M. We give the representations and the group invertibility of M under each of the following conditions:(1)S=0; (2) S is group invertible and CA^πB=0, where A^π=I-AA^#. And the second result generalizes a result of C. Bu et al. [Appl. Math. Comput., 2009, 215: 132–139]

Keywords

Skew field / group inverse / generalized Schur complement / block matrix

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg