Inverse Iteration - Theory and Convergence

Theory and Convergence

The basic idea of the power iteration is choosing an initial vector b (either an eigenvector approximation or a random vector) and iteratively calculating . Except for a set of zero measure, for any initial vector, the result will converge to an eigenvector corresponding to the dominant eigenvalue.

The inverse iteration does the same for the matrix, so it converges to eigenvector corresponding to the dominant eigenvalue of the matrix . Eigenvalues of this matrix are where are eigenvalues of A. The largest of these numbers correspond to the smallest of It is obvious to see that eigenvectors of matrices A and are the same. So:

Conclusion: the method converges to the eigenvector of the matrix A corresponding to the closest eigenvalue to

In particular taking we see that converges to the eigenvector corresponding the smallest in absolute value eigenvalue of A.