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Split $A=D+L+U$ with $D={\mu }^{-1}I$, $L$ the negative unit subdiagonal and $U$ the positive unit superdiagonal. The Jacobi method uses $x^{(r+1)}=-D^{-1}(L+U)x^{(r)}+D^{-1}b$, soThe Gauss-Seidel method uses $(D+L)x^{(r+1)}=b-U{x}^{(r)}$, giving by forward substitutionBoth iterations converge for every initial error exactly when their spectral radius is below one, since the error evolves as powers of the iteration matrix.

For Jacobi, the leading tridiagonal determinant recurrence is $p_j(\lambda )=\lambda p_{j-1}(\lambda )+{\mu }^2{p}_{j-2}(\lambda )$, with $p_0=1,p_1=\lambda$. ThusIts roots are $\lambda =\pm i\mu \sqrt{(3+\sqrt{5})/2}$ and $\pm i\mu \sqrt{(3-\sqrt{5})/2}$. Expanding the Gauss-Seidel determinant givesso its nonzero eigenvalues are $-{\mu }^2(3\pm \sqrt{5})/2$. ThereforeThe skew-tridiagonal Jacobi and Gauss-Seidel iterations consequently have the same convergence range:Equality fails the every-start convergence condition, and larger magnitudes have unstable modes. The nonsymmetric matrix does not permit an unexamined positive-definite-matrix convergence argument.