The Representation and Parametrization of Orthogonal Matrices

Four representations and parametrizations of,orthogonal matt-ices Q is an element of R-mxn in terms of the minimal niimber of essential paraineters {phi} are discussed: the exponential representation, the Householder reflector representation, the Givens rotation representation, and the rational Cayley transform representation. Both square n = m and rectangular fr < m situations are considered. Two separate kinds of parametrizations are considered one in which the individual columns of :Q are distinct; the Stiefel manifold, and the other in which only span(Q) is significant, the Grossmann manifold. The practical issues of numerical stability, continuity, and uniqueness are discussed. The computation of Qin terms of the essential parameters {01, and also the extraction of {phi} for a given Q are Considered for all of the parametrizations. The transformation of gradient arrays between the Qand {phi} variables is discussed for all representations. It is out hope that developers of new methods will benefit from this comparative presentation of an important but rarely analyzed subject.