Proof of theorem about iterative methods

Question

How do I prove that if $A$ is a tridiagonal (or block tridiagonal) matrix then the corresponding $P_J$ and $P_G$ iteration matrices for the Jacobi and Gauss-Seidel methods satisfy that if $\lambda$ is an eigenvalue of $P_J$ then $\sqrt{\lambda}$ (plus or minus?) is one of $P_G$ and if $\lambda$ is an eigenvalue of $P_G$ then $\lambda^2$ is one of $P_J$, and therefore the two methods either both converge or non converges and Gauss-Seidel is twice as fast as Jacobi? This theorem has been given to me in University without demonstration, but then on $A=\left[\begin{smallmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{smallmatrix}\right]$ I obtained the most unexpected result: that Jacobi converged and Gauss-Seidel didn't. Since I can't see any error in the implementation of the algorithms, I can either question this theorem or suppose there is some huge algorithmic error at work, so how do I prove that theorem? And is it true in the first place?