NII Technical Report (NII-2020-004E)：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 minx ∈ Rⁿ \| b - A x \|₂², where A ∈ Rⁿˣⁿ 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(A^T) $ was not complete. In this paper, we will give a complete proof.