A NEW LOOK AT PENCILS OF MATRIX VALUED FUNCTIONS

Matrix pencils depending on a parameter and their canonical forms unde r equivalence are discussed. The study of matrix pencils or generalize d eigenvalue problems is often motivated by applications from linear d ifferential-algebraic equations (DAEs). Based on the Weierstrass-Krone cker canonical form of the underlying matrix pencil, one gets existenc e and uniqueness results for linear constant coefficient DAEs. In orde r to study the solution behavior of linear DAEs with variable coeffici ents 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 canonical forms there, too. Based on these results, we o btain new existence and uniqueness theorems for differential-algebraic systems, which generalize the classical results of Weierstrass and Kr onecker.