Some geometric considerations about the Kronecker normal form

In the first part of this paper we develop general algorithms from the Kronecker normal form of any matrix pencil, and give some of their properties. In the second part these results are applied to the so-called restricted matrix pencil associated with a given (C, A, B) system and introduced by Jaffe and Karcanias (1981). This allows us to make a link with well known algorithms of the geometric approach, particularly the (A, B)-invariant subspace algorithm and the almost (A, B)-controllability subspace algorithm. Finally, relationships between Morse's invariants (Morse 1973) and the invariants of the restricted matrix pencil are exhibited.