Using the shifted power method I find the eigenvalue (of the matrix A) farthest from a number $\mu$ and the corresponding eigenvector .
In the method I follow the below steps:
I first compute the matrix $A-\mu I$ which has the eigenvalues $\{ \lambda_{j}-\mu \}^{n}_{j=1}$ since $Ax = \lambda_{j} x \Leftrightarrow (A-\mu I)x = (\lambda_{j}-\mu)x$
Using the power method on $A-\mu I$ I obtain the eigenvalue farthest from $\mu$ (i.e $\vert \lambda_{j}-\mu \vert$). My question is: Why? I know that the power method find the biggest eigenvalue $\lambda_{j}-\mu$, so how can I conclude that it finds the biggest absolute eigenvalue?