Here's my reasoning. Power iteration converges towards the eigenvector with the largest eigenvalue. Inverse iteration is power iteration using the matrix $(A-\mu I)^{-1}$. The eigenvalues of this matrix are $(\lambda_i – \mu)^{-1}$ with the $\lambda_i$ being the eigenvalues of $A$, so the largest eigenvalue of $(A-\mu I)^{-1}$ is going to correspond to the smallest eigenvalue of $A$. So inverse iteration converges towards the eigenvector with the smallest eigenvalue.
But inverse iteration is supposed to converge towards the eigenvector with the eigenvalue closest to $\mu$. So i made a mistake somewhere. What am i getting wrong ?
Best Answer
If $B = A - \mu I$, iterating $B$ converges (under appropriate conditions) to an eigenvector for the largest eigenvalue (in magnitude) of $B$. Iterating $B^{-1}$ should similarly converge to an eigenvector for the largest eigenvalue of $B^{-1}$. Now the eigenvalues of $B^{-1} = (A - \mu I)^{-1}$ are $(\lambda - \mu)^{-1}$ where $\lambda$ are the eigenvalues of $A$. The absolute value of this is largest when $\lambda$ is closest to $\mu$.