The Calculation of Specified Eigenvectors by Inverse Iteration

Abstract

When an approximation μ is known to an eigenvalue of a matrix A, inverse iteration provides an efficient algorithm for computing the corresponding eigenvector. It consists essentially of the determination of a sequence of vectors x _r defined by

$$ \matrix{ {(A - \mu I){x_{r + 1}} = {k_r}{x_r}} & {(r = 0,1, \ldots ),} \cr } $$

((1))

where k _r is chosen so that ‖x _r+1‖ = l in some norm and x ₀ is an arbitrary unit vector.