# The Quasi-Weierstraß form for regular matrix pencils

Dedicated to Heinrich Voß on the occasion of his 65th birthday

Regular linear matrix pencils A - E \partial \in \mathbb{K}^{n \partial n} [\partial], where \mathbb{K} = \mathbb{Q}, \mathbb{R} or \mathbb{C}, and the associated differential algebraic equation (DAE) E\dot{x} = Ax are studied. The Wong sequences of subspaces are tackled and invoked to decompose the \mathbb{K}^n into \mathcal{V}^* \oplus \mathcal{W}^*, where any bases of the linear spaces \mathcal{V}^* and \mathcal{W}^* transform the matrix pencil into the Quasi-Weierstraß form. The Quasi-Weierstraß form of the matrix pencil decouples the original DAE into the underlying ODE and the pure DAE or, in other words, decouples the set of initial values into the set of consistent initial values \mathcal{V}^* and "pure" inconsistent initial values \mathcal{W}^* \ \{0\} . Furthermore, \mathcal{V}^* and \mathcal{W}^* are spanned by the generalized eigenvectors at the finite and infinite eigenvalues, resp. The Quasi-Weierstraß form is used to show how chains of generalized eigenvectors at finite and infinite eigenvalues of A − E \partial lead to the well-known Weierstraß form. So the latter can be viewed as a generalized Jordan form. Finally, it is shown how eigenvector chains constitute a basis for the solution space of E \dot{x} = Ax.

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Berger, Thomas / Ilchmann, Achim / Trenn, Stephan: The Quasi-Weierstraß form for regular matrix pencils. 2009.

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