Abstract
In this paper, we introduce a general class of quasi-sparse potential companion pencils for arbitrary square matrix polynomials over an arbitrary field, which extends the class introduced in [B. Eastman, I.-J. Kim, B. L. Shader, K.N. Vander Meulen, Companion matrix patterns. Linear Algebra Appl. 436 (2014) 255–272] for monic scalar polynomials. We provide a canonical form, up to permutation, for companion pencils in this class. We also relate these companion pencils with other relevant families of companion linearizations known so far. Finally, we determine the number of different sparse companion pencils in the class, up to permutation.
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Acknowledgements
We wish to thank two anonymous referees for the careful reading of this paper and for their comments that allowed us to improve the presentation.
This work has been partially supported by the Ministerio de Economía y Competitividad of Spain through grants MTM2015-68805-REDT and MTM2015-65798-P.
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Terán, F.D., Hernando, C. (2019). A Class of Quasi-Sparse Companion Pencils. In: Bini, D., Di Benedetto, F., Tyrtyshnikov, E., Van Barel, M. (eds) Structured Matrices in Numerical Linear Algebra. Springer INdAM Series, vol 30. Springer, Cham. https://doi.org/10.1007/978-3-030-04088-8_8
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