On Variations of Power Iteration Seungjin Choi Department of Computer Science Pohang University of Science and Technology San 31 Hyoja-dong, Nam-gu Pohang 790-784, Korea seungjin@postech.ac.kr Abstract. The power iteration is a classical method for computing the eigenvector associated with the largest eigenvalue of a matrix. The sub- space iteration is an extension of the power iteration where the subspace spanned by n largest eigenvectors of a matrix, is determined. The nat- ural power iteration is an exemplary instance of the subspace iteration, providing a general framework for many principal subspace algorithms. In this paper we present variations of the natural power iteration, where n largest eigenvectors of a symmetric matrix without rotation ambiguity are determined, whereas the subspace iteration or the natural power it- eration finds an invariant subspace (consisting of rotated eigenvectors). The resulting method is referred to as constrained natural power itera- tion and its fixed point analysis is given. Numerical experiments confirm the validity of our algorithm. 1 Introduction A symmetric eigenvalue problem where the eigenvectors of a symmetric matrix are required to be computed, is a fundamental problem encountered in a va- riety of applications involving the spectral decomposition. The power iteration is a classical and the simplest method for computing the eigenvector with the largest modulus. The subspace iteration is a natural generalization of the power iteration, where the subspace spanned by n largest eigenvectors of a matrix, is determined. The natural power iteration [1] is an exemplary instance of the subspace iteration, that was investigated mainly for principal subspace analysis. In this paper we present variations of the natural power iteration and show that its fixed point is the n largest eigenvectors of a symmetric matrix up to a sign ambiguity, whereas the natural power iteration just finds a principal subspace (i.e., arbitrarily rotated eigenvectors). The resulting algorithm is referred to as constrained natural power iteration. Numerical experiments confirm the validity of our algorithm. 2 Natural Power Iteration The power iteration is a classical method which finds the largest eigenvector (associated with the largest eigenvalue) of a matrix C ∈ R m×m [2]. Given a