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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?

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  • $\begingroup$ I don't know of any method in the general case… $\endgroup$
    – Dirk
    Apr 20 at 12:23

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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 $$\rho(I-\omega A) < 1 \text{ if and only if } 0 < \omega < \frac{2}{\lambda_\max(A)}.$$

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