  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 $ \min_{ {\bf x} \in {\bf R}^n}
{\| {\bf b} - A {\bf x} \|_2}^2$, where $ A \in {\bf R}^{n \times n}$ may be
singular and $ {\bf b} \in {\bf R}^n$, by decomposing the algorithm into the
range $ {\cal R}(A) $ and its orthogonal complement $ {\cal R}(A)^\perp $
components. However, we found that the proof of the fact that GMRES gives a
least squares solution if $ {\cal R}(A) = {\cal R}(A^{\scriptsize T} ) $ was
not complete. In this paper, we will give a complete proof.
