GMRES on singular systems revisited

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.