[Math] Shifter Power Method

Question

If we know all the eigenvalues of a matrix A except the largest one. We want to apply shifted power iteration to get the largest eigenvalue. Something like $(A-\alpha I)$ . Then what should be the value of shift ($\alpha$) to make it converge as fast as possible ?

Answer 1

If the eigenvalues of $A$ are all real:

$$\lambda_1 \gt \lambda_2 \gt \ldots \gt \lambda_k$$

then the power iterations of the shifted matrix $A - \alpha I$ converge at the optimal rate when $\alpha = \frac{\lambda_2 + \lambda_k}{2}$.

Something similar can be done if the eigenvalues are complex but known except for the one of greatest modulus.