I am studying for my exam in numerical mathematics and I was thinking about the relationship between the eigenvalues of a matrix in a linear system and the corresponding iteration matrix. So let $A\in\mathbb{R}^{n\times n},$ $b\in\mathbb{R}^n$ and we want to solve the linear system $Ax=b$.
Let $A=D-E-F$ be composition in its diagonal and a left/right upper diagonal matrix. Then the iteration matrices for the (damped) Jacobi- resp. Gauss-Seidel method are $K^{J}=I-\omega D^{-1}A$ (if $\omega=1$ it is the classical, undamped Jacobi-method) resp. $K^{GS}=I-(D-E)^{-1}A$ for Gauss-Seidel.
Now assume we already know the eigenvalues of $A$. Is there any chance to determine the eigenvalues of the iteration matrices and also their spectral radii?
Usually you check convergence by the theory for symmetric positive definite or (strictly) diagonaldominant matrices.
Using the Eigenvalues of the system matrix $A$, I am aware of the following relation: For the damped Jacobi method with symmetric positive definite matrix $A$, you have the relation $$\rho(I-\omega A) < 1 \text{ if and only if } 0 < \omega < \frac{2}{\lambda_\max(A)}.$$