= Solution
A sequence is <Q-linear convergence>[Q-linearly convergent] to $x^*$ in a specified norm if $\|x_{k+1}-x^*\|\leq q\|x_k-x^*\|$ for all sufficiently large $k$, with some $q<1$. Its asymptotic factor is the limsup of consecutive error ratios.
Put $h=c^{-2}+d^{-2}$, $b_0=c^{-2}-d^{-2}$ and $t=b_0/h$. For an error $e=x-x^*$, an exact first-coordinate step sets $e_1=-te_2$, and an exact second-coordinate step sets $e_2=-te_1$. Once the first coordinate step has occurred, errors consequently alternate between the two lines
$$
e=(-tz,z),\qquad e=(z,-tz).
$$
The next error has its free coordinate multiplied in magnitude by $|t|$, and both lines have the same Euclidean norm factor $\sqrt{1+t^2}$. Therefore, until finite termination,
$$
\boxed{\frac{\|e_{k+1}\|_2}{\|e_k\|_2}
=|t|=\frac{|c^2-d^2|}{c^2+d^2}
=\frac{\kappa_2(H)-1}{\kappa_2(H)+1}<1.}
$$
This is a sharp bound, attained after the first step; the objective errors have factor $t^2$ per step. If $c^2=d^2$, the problem separates and at most two exact steps suffice. When the <condition number> is large, the factor approaches one: the iterates zigzag slowly across a narrow tilted valley.
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