In matrix form the Gauss Seidel iterations can be expressed as:
$$X^{m+1} = (D + L)^{-1}\textbf{b} - (D + L)^{-1}UX^{m}$$
Where $A = D + L +U$ is a finite difference matrix expressing the original discretized differential equation. And $D, L, U$ are a diagonal, lower diagonal, upper diagonal matrices.
I need to find the eigenvalues of $(D+L)^{-1}U$ I have some shortcuts. I know that $(D + L)^{-1}$ is an upper triangular matrix and thus $(D+L)^{-1}U$ is also an upper diagonal matrix.
If we assume $A$ was a central difference matrix then we also know that $D$ is constant, i.e. it can be written as $aI$ for some real value $a$. $L = U^T$ have a single constant coefficient along the off main diagonal diagonals and 0 everywhere else.
So we would have $(aI + U^T)U$
Can I do anything to express the eigenvalues based on the entries of $A$ from this point?