  The techniques and analysis presented in this thesis provide new methods to
solve optimization problems posed on Riemannian manifolds. These methods are
applied to the subspace tracking problem found in adaptive signal processing
and adaptive control. A new point of view is offered for the constrained
optimization problem. Some classical optimization techniques on Euclidean space
are generalized to Riemannian manifolds. Several algorithms are presented and
their convergence properties are analyzed employing the Riemannian structure of
the manifold. Specifically, two new algorithms, which can be thought of as
Newton's method and the conjugate gradient method on Riemannian manifolds, are
presented and shown to possess quadratic and superlinear convergence,
respectively. These methods are applied to several eigenvalue and singular
value problems, which are posed as constrained optimization problems. ...
