Extending the Notions of Companion and Innnite Companion to Matrix Polynomials Extending the Notions of Companion and Innnite Companion to Matrix Polynomials

An extended innnite companion matrix ~ C 1 (D) and an innnite companion matrix C 1 (D) for a (nonmonic in general) matrix polynomial D is introduced and the nite companion matrix C(D) is generalized to the nonmonic case. These matrices generalize all properties of the innnite and nite companion (Frobenius) matrix corresponding to a scalar polynomial. In particular, C 1 (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 (deened on vector polynomials) to the remainder subspace corresponding to D, with characteristic polynomial equal to det D. A factorization formula for nite-rank block Hankel matrices is proved. The generalization of the nite 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 diierence or diierential equations with constant coeecients can be transformed into a rst order system of dimension n = det D. Extending the notions of companion and innnite companion to matrix polynomials Abstract An extended innnite companion matrix ~ C1(D) and an innnite companion matrix C1(D) for a (nonmonic in general) matrix polynomial D is introduced and the nite companion matrix C(D) is generalized to the nonmonic case. These matrices generalize all properties of the innnite and nite companion (Frobenius) matrix corresponding to a scalar polynomial. In particular, C1(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 (deened on vector polynomials) to the remainder subspace corresponding to D, with characteristic polynomial equal to det D. A factorization formula for nite-rank block Hankel matrices is proved. The generalization of the nite 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 diierence or diierential equations with constant coeecients can be transformed into a rst order system of dimension n = det D.