Solution
= Solution
Writing the error as $e_k=x_k-x^*$ gives
$$
e_{k+1}=He_k
$$
provided the fixed-point consistency condition
$$
x^*=Hx^*+v
$$
holds. Convergence for every starting <vector> is therefore equivalent to $H^k\to0$, which in finite dimensions is equivalent to
$$
\rho(H)<1.
$$
These two conditions are necessary and sufficient.