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TW 261
M. Van Barel, V. Pták and Z. Vavrín
Extending the notions of companion and infinite companion to matrix polynomials
Abstract
An extended infinite companion matrix C~∞(D) and an infinite companion matrix C∞(D) for a (nonmonic in general) matrix polynomial D is introduced and the finite companion matrix C(D) is generalized to the nonmonic case. These matrices generalize all properties of the infinite and finite companion (Frobenius) matrix corresponding to a scalar polynomial. In particular, C∞(D) is a controllability matrix of a system whose inner behaviour is given by D, and C(D) is a compression of the shift operator (defined on vector polynomials) to the remainder subspace corresponding to D, with characteristic polynomial equal to det D. A factorization formula for finite-rank block Hankel matrices is proved. The generalization of the finite companion matrix C(D) permits to construct new linearizations of nonmonic matrix polynomials. These linearizations have considerably smaller dimension than the standard ones. As a consequence, any system of linear difference or differential equations with constant coefficients can be transformed into a first order system of dimension n = det D.
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