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 coefficient DAEs.

In order to study the solution behaviour 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 canonical forms 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.
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