= Solution
Here $D=I$, so the Jacobi iteration matrix is
$$
H=I-A=
\begin{pmatrix}
0&-\alpha&-\alpha\\
-\alpha&0&-\alpha\\
-\alpha&-\alpha&0
\end{pmatrix}.
$$
The all-ones vector is an <eigenvector> with <eigenvalue> $-2\alpha$. On its two-dimensional <orthogonal complement>, the coordinates sum to zero and $Hv=\alpha v$, so the other eigenvalue is $\alpha$ with multiplicity two. Therefore
$$
\rho(H)=\max\{2|\alpha|,|\alpha|\}=2|\alpha|.
$$
By <Jacobi convergence for a three-by-three equicorrelation matrix>, convergence occurs exactly for
$$
\boxed{-\frac12<\alpha<\frac12.}
$$
At either endpoint the spectral radius is one, so convergence for arbitrary initial data fails.
Back to article page
