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Pseudo Principal Pivot Transform: The Group Inverse Case

In this short note, we prove a formula for the group inverse of a block matrix and consider the pseudo principal pivot transform expressed in terms of group inverses. Extensions of the usual principal pivot transform, where the usual inverse is replaced by the Moore-Penrose inverse, were considered in the literature. The objective here is to derive the corresponding formulae for the group generalized inverse. These are expected to be useful in our future work.