To solve a linear system $Ax=b$ using the Gauss-Seidel method, the matrix $A$ can be decomposed as $A = L + D + U$ and solved iteratively using
$$ x^{(\text{new})} = -(D + L)^{-1} U x^{(\text{old})} + (D + L)^{-1} b $$
Successive over-relaxation (SOR) adds a relaxation parameter $\omega \in [0,1]$ and solves the system using
$$ x^{(\omega)} = (1-\omega) x^{(\text{old})} + \omega x^{(\text{new})} $$
I saw a lecture that stated:
The relaxation parameter acts like an effective increase in the eigenvalues of the matrix. A small enough value can enable convergence.
Can someone explain that statement to me? How does the relaxation parameter increase the eigenvalues and how does it affect convergence?