Gauss-Seidel iteration can diverge for an invertible matrix ($\rho (H)=4$)

An invertible matrix and nonzero diagonal entries make the update well defined, but do not guarantee convergence. For the original linear systemthe Gauss-Seidel method updates $x^{(k+1)}=1-2y^{(k)}$ and $y^{(k+1)}=1-2x^{(k+1)}=-1+4y^{(k)}$. Starting at zero gives $(1,-1)$, then $(3,-5)$. The second-coordinate forward error is $e_y^{(k)}=-4^k/3$, and the iteration matrix $H=\left(\begin{array}{cc}0& -2\\ 0& 4\end{array}\right)$ has spectral radius $4$. The linear system has a unique solution while this particular stationary iterative method for a linear system diverges.