 
Summary: 07 Apr 2008 Tech. Report UCLINMA2008.013
A geometric Newton method for Oja's vector field
P.A. Absil
M. Ishteva
L. De Lathauwer§
S. Van Huffel
April 7, 2008
Abstract
Newton's method for solving the matrix equation F(X) AX 
XXT AX = 0 runs up against the fact that its zeros are not isolated.
This is due to a symmetry of F by the action of the orthogonal group.
We show how differentialgeometric techniques can be exploited to re
move this symmetry and obtain a "geometric" Newton algorithm that
finds the zeros of F. The geometric Newton method does not suffer
from the degeneracy issue that stands in the way of the original Newton
method.
Key words. Oja's learning equation, Oja's flow, differentialgeometric optimization, Rie
mannian optimization, quotient manifold, neural networks
1 Introduction
Let A be a symmetric positivedefinite n × n matrix and let p be a positive integer smaller
