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In this paper, we revisit the core inverse introduced by Baksalary and Trenkler. We first give some new characterizations of the core inverse. Then, we give a new representation of the core inverse, which is related to A T,S (2) .
In this paper, some conditions for the nonsingularity and group inverses of linear combinations of generalized and hypergeneralized projectors are established. Moreover, some formulae for the inverses and group inverses of them are derived. The work of this paper extends some previous results.
The invertibility of combinations of two orthogonal projectors a P + b Q + c PQ + d QP is researched by using the CS-decomposition of matrices and properties of orthogonal projectors. The Moore-Penrose inverse of the combinations is presented under some special conditions.