= Solution
Subtract the exact solution from the <Jacobi method> to get $e_n=He_{n-1}$, hence $e_n=H^ne_0$. Convergence from every initial vector is equivalent to $H^n\to0$. If every <eigenvalue> has modulus below one, a <Jordan block> $\lambda I+N$ has powers that are finite sums $\binom nk\lambda^{n-k}N^k$. The <polynomial> factor in $n$ is dominated by geometric decay; for $\lambda=0$, the block is nilpotent. Consequently $H^n\to0$.
Conversely, for a complex <eigenvector> $v$ with <eigenvalue> $\lambda$, $H^nv=\lambda^nv$. Convergence for real errors also implies convergence for their complexification, so this tends to zero only if $|\lambda|<1$. Therefore \b[the iteration converges from every starting vector if and only if $\rho(H)<1$], where $\rho$ is the <spectral radius>.
For the specified <matrix>,
$$
H=\begin{pmatrix}0&0&\mu\\\mu/3&0&\mu/3\\\mu&0&0\end{pmatrix},\qquad \det(tI-H)=t(t^2-\mu^2).
$$
Its <eigenvalues> are $0,\mu,-\mu$, so $\rho(H)=|\mu|$. Hence \b[$-1<\mu<1$] is the convergence range. The original coefficient <matrix> has determinant $12(1-\mu^2)$, so the endpoints also violate its assumed nonsingularity.
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