Representation and approximation of the outer inverse \(A_{T,S}^{(2)}\) of a matrix \(A\) (Q1977374)

The authors establish a basic representation and a representation theorem for the outer inverse \(A^{(2)}_{T,S}\) of a matrix \(A\), which is the matrix \(X\) satisfying \[ XAX=X,\;R(X)=T\text{ and }N(X)=S. \] Several specific representations and iterative methods for \(A^{(2)}_{T,S}\) are given. The paper shows that this representation includes many of the traditional generalized inverses and outer inverses, and the relation of the results to these outer inverses will be explored.