# Multigrid: Gauss–Seidel eigenvalues and eigenvectors

I am trying to figure out the questions from Multigrid Tutorial by Briggs. However I am stuck on these two questions.

https://www.researchgate.net/publication/220690328_A_Multigrid_Tutorial_2nd_Edition

(a) Show that the eigenvalue problem for the Gauss-Seidel iteration matrix, $$R_{G} \mathbf{w}=\lambda \mathbf{w}$$, may be expressed in the form $$U \mathbf{w}=(D-L) \lambda I \mathbf{w}$$, where $$U, L, D$$ are defined in the text.

(c) Show that the eigenvector and eigenvalue are associated with $$\lambda_{k}$$ = $$\cos ^{2}\left(\frac{k \pi}{N+1}\right)$$ is $$w_{k, j}=\left[\cos \left(\frac{k \pi}{N+1}\right)\right]^j \sin \left(\frac{j k \pi}{N+1}\right)$$.

My attempt

For part a, I derived $$R_{G}=(D-L)^{-1} U$$ where D, L, U is the diagonal matrix of A. I am not sure how to get the answer as shown in question (a).

I am lost to show the eigenvector and eigenvalue relationship for part c.