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


(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.


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