TOPOLOGICAL ASPECTS OF THE PARTIAL-REALIZATION PROBLEM

In this paper we study the topology of manifolds of rectangular Hankel matrices, motivated by the problem of partial realization. Following Fischer and Frobenius, we introduce a rank-preserving G1(2, R)-action on the space Hank(M x N) of all real M x N Hankel matrices. We derive an explicit formula for the first n x n principal minor of transformed Hankel matrices. Extending the earlier work of Brockett, the formula is applied to introduce a manifold structure on the space Hank(n, M x N) of all M x N Hankels of rank n. We construct a cell decomposition of Hank(M x N) which induces a cellular subdivision on each of the manifolds Hank(n, M x N) where n less-than-or-equal-to min(M, N). This new cell decomposition is applied to investigate the topology of partial realizations.