Inverse Iteration
Inverse Iteration is a method for finding an eigenvector of a matrix $A$ for a known approximate eigenvalue $m$ :
- it’s Power Iteration applied to $(A - m I)^{-1}$
- at each step the system $(A - m I) \, \mathbf x_{k+1} = \mathbf x_k$ is solved, e.g. with LU Decomposition, which can be computed once and then reused
- the eigenvalue of $(A - m I)^{-1}$ that has the largest magnitude is $1 / (\lambda - m)$, so the method converges to the eigenvector of the eigenvalue $\lambda$ closest to $m$
It’s used in the QR Algorithm for computing the eigenvectors once the eigenvalues are approximated.