A new vector field method for eigen-decomposition of symmetric matrices

We study a new approach to eigen-decomposition of symmetric matrices, called the vector field method. In contrast to the methods currently in use, the vector field method computes an eigenvector by computing a discrete approximation to the integral curve of a special tangent vector field on the unit sphere. The optimization problems embedded in each iteration of the vector field algorithms admit closed-form solutions making the vector field approach promisingly efficient.Besides establishing the general convergence results about the vector field method and the complexity bounds for computing ϵϵ-approximations to eigenvectors, we also computationally compare the performance of a family of algorithms called the recursive vector field algorithms with that of the power method. The vector field method, like the power method, offers a significant advantage in that it can exploit sparseness to speed up computation even further. The convergence analysis, that we present, also suggests that one could speed up the vector field algorithms through preprocessing that contracts the spectral radius of the given matrix.