  We study the problem of approximating orthogonal matrices so that their
application is numerically fast and yet accurate. We find an approximation by
solving an optimization problem over a set of structured matrices, that we call
extended orthogonal Givens transformations, including Givens rotations as a
special case. We propose an efficient greedy algorithm to solve such a problem
and show that it strikes a balance between approximation accuracy and speed of
computation. The approach is relevant to spectral methods and we illustrate its
application to PCA.
