 
Summary: Submitted to Foundations of Computational Mathematics on 23 Mar 2004
Wellposedness and regularity properties of the
GrassmannRayleigh quotient iteration
P.A. Absil
U. Helmke
K. H¨uper
Abstract. A generalization of the Rayleigh quotient iteration has recently been proposed
on the Grassmann manifold. This iteration has been shown to converge locally cubically to
the invariant subspaces of symmetric matrices. The present paper studies global properties
of the iteration mapping. Results are obtained e.g. concerning fixed points, smoothness, and
singularities of the iteration mapping.
Key words. GrassmannRayleigh quotient iteration, BlockRayleigh quotient iteration,
Grassmann manifold, singularities, continuous extension, fixed points.
AMS subject classification. 65F15.
1 Introduction
The Rayleigh quotient iteration (RQI) is a wellknown method for computing an eigenvector
of a symmetric matrix. It consists in a shifted inverse iteration that maps y Rn\{0} to y+
defined by
(A  (y)I)z = y, y+ = µz (1)
where µ is a normalization factor and (y) := (yT Ay)/(yT y) is the Rayleigh quotient of A
