Will the GMRES iterative method for finding solutions to systems of equations, $Ax=b$, always converge after one iteration if the initial guess vector $b$ ($q_1 = b / \|b\|$) is an eigenvector of $A$? I think this is true, but I can't seem to make a strong statement about it, but what I have so far is.
Given an eigenvector of $A$, $b$, at the first iteration, well even before the first iteration, of the Arnoldi iteration we have $q_1 = b / \|b\|$. If we then take the Ritz value, $\theta_1 = q_1^T A q_1$, at this first iteration we find that it is equivalent to the Rayleigh quotient at an eigenvector of $A$, which produces the correspodning eigenvalue of $b$, $\lambda_b$.
Next, in GMRES we attempt to find $x_n \in K_n$ where $K_n$, the $n$th Krylov subspace, is produced from the vectors $q_j$ of the Arnoldi iteration. This potential solution vector $x_n$ can be written in terms of these $q_j$ as $x_n = Q_n y_n$ (where $y_n \in \Re^n)$.
At this first iteration we know $Q_n$ contains only $q_1$ which is a normalized eigenvector of $A$. Thus we have $x_1 = q_1 y_1$. If we allow $y_1 = 1 / \lambda_b$ then $x_n$ is a solution to $Ax=b$. (We allow $y_1$ to equal this because a solution to $Ax=b$ when $b$ is an eigenvector of $A$ is $x = \frac{1}{\lambda} b$).
I think this makes sense, to me at least, but again it seems kind of weak. Specifically it doesn't address how long Arnoldi is ran before GMRES commences. I want to say $0$ iterations (so that $Q_1$ only contains $q_1$), but I'm not sure if that is necessary to enforce. It also hinges on that fact that we can "arbitrarily" select the value of $y_1$, though, if $b$ is the eigenvector shouldn't $y_1$ be selected as this value to minimize the residual?
Couldn't a separate, much simpler, proof be made stating that the solution vector lies in the $K_1$ subspace because $x = \frac 1 \lambda b$ lies in $K_1 = \operatorname{span} \lbrace b \rbrace$?