= Solution
One restarted step chooses $x_j=x_{j-1}+\alpha_jr_{j-1}$ to minimize the new residual norm. Since $r_{j-1}=A(x-x_{j-1})$,
$$
x-x_j=(I-\alpha_jA)(x-x_{j-1}),
$$
and induction gives
$$
\boxed{x-x_k=q_k(A)\cdots q_1(A)(x-x_0),\qquad q_j(t)=1-\alpha_jt.}
$$
These polynomials have degree at most one; a zero step gives the constant polynomial one.
In the specified example the first search space is spanned by $b=(1,1)^T$. Writing $x_1=\alpha b$, its squared residual norm is $(1-\alpha)^2+(1+\alpha)^2=2+2\alpha^2$, uniquely minimized at $\alpha=0$. Thus restarted <GMRES> gives $\boxed{x_1=x_2=(0,0)^T}$ and stagnates forever, repeatedly facing the same minimization problem.
Without restarting, the first iterate is still zero, but $K_2(A,b)=\operatorname{span}\{(1,1)^T,(1,-1)^T\}=\mathbb R^2$. The second minimization finds the exact solution $\boxed{x_2=(1,-1)^T}$. This <one-step restarted GMRES stagnation example> shows how discarding the accumulated search space can prevent convergence even for a nonsingular two-dimensional system.
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