A new class of g-inverse of square matrices
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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(A 2 ). The following properties of A - C are established: (a) A - C = A(A 2 ) - . (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 (A 2 ) - =(A 3 )-A leads to A - C =A(A 3 )-A which is unique irrespective of the choice of (A 3 ) - and is, in fact, the same as the Scroggs-Odell pseudoinverse (J.SIAM 1966) of A. When R(A)=R(A 2 ), this indeed is a much simpler way of calculating the Scroggs-Odell pseudoinverse compared to the method indicated by its authors. (e) A(A 3 )-A belongs to the subalgebra generated by A.
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[2] The commuting inverses of a square matrix , 1966 .