Abstract 
In [Hayami K, Sugihara M. Numer Linear Algebra Appl. 2011; 18:449469], the authors analyzed the convergence behaviour of the Generalized Minimal Residual (GMRES) method for the least squares problem $ \min_{x \in R^n} {\ b  A x \_2}^2$, where $ A \in R^{nxn}$ may be singular and $ b \in R^n, by decomposing the algorithm into the range $ R(A) $ and its orthogonal complement $ R(A)^\perp $ components. However, we found that the proof of the fact that GMRES gives a least squares solution if $ R(A) = R(A^T) $ was not complete. In this paper, we will give a complete proof.
