 Skew-tridiagonal Jacobi and Gauss-Seidel iterations

For $A={\mu }^{-1}I+U-U^T$, with $U$ the four-dimensional unit superdiagonal, the Jacobi method has eigenvalues $2i\mu \cos (j\pi /5)$, $j=1,\dots ,4$. The Gauss-Seidel method has two zero eigenvalues and $-{\mu }^2(3\pm \sqrt{5})/2$. Therefore both methods converge for every starting vector exactly when $0<|\mu |<(\sqrt{5}-1)/2$. This equality of convergence ranges follows from their spectra, not from positive-definiteness of the nonsymmetric coefficient matrix.