Overlapping parametrizations for spaces of orthogonal matrices

As is well-known, the space of orthogonal matrices of a given size possesses a nontrivial manifold structure and is not homeomorphic to Euclidean space. If one is dealing with an optimization problem concerning such a space this will inevitably locally cause numerical illconditioning problems when a classical approach is followed where a Euclidean parametrization for a generic subspace is used. These problems can be avoided, though, by a di erentialgeometric approach, provided an atlas of overlapping local coordinate charts is available in conjunction with a chart selection mechanism. This is one motivation for an explicit construction of such atlases. Orthogonal matrices are also intimately related with representations of discrete-time stable allpass systems (both in the scalar and multivariable case) and appear in the construction of canonical forms for classes of such systems. Atlases for spaces of orthogonal matrices make it possible to parametrize these classes of systems in an overlapping balanced way, thus enabling geometric methods for system identi cation and model reduction. This report deals with the explicit construction of atlases of overlapping parametrizations for several di erent spaces of orthogonal matrices. In particular we address the space of p m orthogonal matrices (of which the space of square p p orthogonal matrices forms a special case), the space of m (m+ n) orthogonal matrices with a xed given rank structure index of order n, and the space of m (m+n) orthogonal matrices with an unspeci ed rank structure index of xed order n. The rst space is of general interest; the latter two are of importance in relation to the topic of discrete-time stable multivariable allpass systems. It is indicated how local parameters for a given orthogonal matrix with respect to a speci ed local coordinate chart can be determined, as well as how all orthogonal matrices in a speci ed coordinate chart can be parametrized.