Matrix pencil approach to geometric system theory

A number of relationships between the geometric and the algebraic linear system theory are briefly surveyed, which may be discussed in terms of the classical theory of matrix pencils. The input-output pencil is defined and used for the characterisations of the geometrical concepts of (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 analysing 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.