A new class of g-inverse of square matrices

Abstract

Necessary and sufficient conditions are obtained for a matrix A to have a g-inverse with rows and columns belonging to special linear manifolds. For a square matrix A, a g-inverse, with columns belonging to the linear manifold generated by the columns of A, is denoted by A^-_C. Such a g-inverse exists if and only if R(A)=R(A²). The following properties of A^-_C are established: (a) A^-_C= A(A²)^-. (b) For any positive integer m, (A^-_C)^m provides a reflexive g-inverse of A^m. (c) If x is an eigenvector corresponding to a nonnull eigenvalue λ of A, x is also an eigenvector of A^-_C corresponding to its eigenvalue 1/λ. The converse of this result is also true. (d) A special choice of (A²)^-=(A³)-A leads to A^-_C=A(A³)-A which is unique irrespective of the choice of (A³)^- and is, in fact, the same as the Scroggs-Odell pseudoinverse (J.SIAM 1966) of A. When R(A)=R(A²), this indeed is a much simpler way of calculating the Scroggs-Odell pseudoinverse compared to the method indicated by its authors. (e) A(A³)-A belongs to the subalgebra generated by A.