On fischer-frobenius transformations and the structure of rectangular block hankel matrices
Wilfried Manthey, D. Hinrichsen, Uwe Helmke · Linear and Multilinear Algebra · 1996
In this paper, we develop three essential ingredients of an algebraic structure theory of finite block Hankel matrices. The development centers around a transformation of block Hankel matrices, first introduced by Fischer and Frobenius for scalar Hankel matrices. We prove three results:First, Iohvidov's fundamental notion of the characteristic of a Hankel matrix is extended to the block matrix case and the relationship between rank and characteristic is clarified. The characteristic of block Hankel matrices involves the introduction of two new sets of invariants, the principal and residual observability and controllability indices. Second, the class of generalized Fischer-Frobenius transformations is shown to be the largest class of rank preserving transformations X↦S T XT which leave the set of block Hankel matrices invariant. Finally the principal minor lemma from scalar Hankel matrix theory is extended to the block matrix case. We use this generalization in order to prove that the set of fixed rank block Hankel matrices is a smooth manifold and establish some elementary topological properties.