In Hayami K, Sugihara M. Numer Linear Algebra Appl. 2011; 18: 449--469, the authors analyzed the convergence behaviour of the Generalized Minimal Residual (GMRES) method for the least squares problem ₗ ₑ䂞 \| b - A x \|₂², where A R^nxn may be singular and b Rⁿ, by decomposing the algorithm into the range R (A) and its orthogonal complement R (A) ^ components. However, we found that the proof of the fact that GMRES gives a least squares solution if R (A) = R (AT) was not complete. In this paper, we will give a complete proof.
Hayami et al. (Thu,) studied this question.