A splitting iterative method for α− β generalized inverse and singular linear system

Abstract

This paper deals with the case when the α− β generalized inverse A α, β (−1) is a linear transformation, in which case we give a splitting iterative method for A α, β (−1). We also show that in the case of linear transformation, the α− β generalized inverse A α, β (−1) is a {1,2}-inverse of the matrix A with prescribed range and null space, based on which we propose a iterative method of calculating the unique α-approximate solution of minimal β-norm of the system Ax= b. The results extend some previous results about the Moore–Penrose inverse A + and the weighted Moore–Penrose inverse A M, N +.