Simple exponential acceleration of the power iteration algorithm

Abstract

Many real-world problems rely on finding eigenvalues and eigenvectors of a matrix. The power iteration algorithm is a simple method for determining the largest eigenvalue and associated eigenvector of a general matrix. This algorithm relies on the idea that repeated multiplication of a randomly chosen vector x by the matrix A gradually amplifies the component of the vector along the eigenvector of the largest eigenvalue of A while suppressing all other components. Unfortunately, the power iteration algorithm may demonstrate slow convergence. In this report, we demonstrate an exponential speed up in convergence of the power iteration algorithm with only a polynomial increase in computation by taking advantage of the commutativity of matrix multiplication.