Polynomials satisfied by two linked matrices

Polynomials in two variables, evaluated at A and with A being a square complex matrix and being its transform belonging to the set {A=, A†, A∗}, in which A=, A†, and A∗ denote, respectively, any reflexive generalized inverse, the Moore–Penrose inverse, and the conjugate transpose of A, are considered. An essential role, in characterizing when such polynomials are satisfied by two matrices linked as above, is played by the condition that the column space of A is the column space of . The results given unify a number of prior, isolated results.