A Note on Explicit Representations of (B, C)-Inverses as Inverses Along Suitable Matrices

In this note, starting from the known relation between (B,C)-inverses and inverses along suitable matrices, we study explicit representations of (B,C)-inverses of complex matrices as inverses along suitable matrices. More precisely, for a given (B,C)-invertible matrix, we investigate explicit choices of a matrix D for which A(B,C)=A‖D holds. For the three natural choices D=B, D=C and D=BC, we give necessary and sufficient conditions under which A(B,C)=A‖D. Also, several recently studied generalized inverses, admitting (B,C)-inverse representations, are represented as inverses along suitable matrices. Finally, using fixed full-rank factorizations of B and C, we explicitly parametrize all matrices D satisfying A(B,C)=A‖D.