MATRIX PENCIL APPROACH TO GEOMETRIC SYSTEM THEORY

A number of relationships between the geometric and the algebraic linear system theory are surveyed, which may be discussed in terms of the classical theory of matrix pencils. The input-output pencil is defined and used for the characterizations of the geometrical concepts (A,B)-invariant subspace, controllability subspace and transmission subspace. The problem of finding the maximal (A,B)-invariant and maximal controllability subspaces contained in another subspace is finally reduced to a problem of analyzing the structure of a particular pencil, the restriction pencil. A common theme running through all the analyses is the use of the canonical forms of Weierstrass and Kronecker.