A new look at pencils of matrix valued functions

Abstract

Matrix pencils depending on a parameter and their canonical forms under equivalence are discussed. The study of matrix pencils or generalized eigenvalue problems is often motivated by applications from linear differential-algebraic equations (DAEs). Based on the Weierstrass-Kronecker canonical form of the underlying matrix pencil, one gets existence and uniqueness results for linear constant coefficients DAEs. In order to study the solution behavior of linear DAEs with variable coefficients one has to look at new types of equivalence transformations. This then leads to new canonical forms and new invariances for pencils of matrix valued functions. We give a survey of recent results for square pencils and extend these results to nonsquare pencils. Furthermore we partially extend the results for canonical forms of Hermitian pencils and give new canonicalforms there, too. Based on these results, we obtain new existence and uniqueness theorems for differential-algebraic systems, which generalize the classical results of Weierstrass and Kronecker.