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Companion matrices and Golden-Fibonacci sequences

  • M. Mousavi
  • M. EsmaeiliEmail author
  • A. Zaghian
Original Research
  • 3 Downloads

Abstract

We assign a companion sequence to a given companion matrix. Given a companion matrix \(\mathbf{C}\), we make use of the associated companion sequence to provide a simple closed-form expression for \(\mathbf{C}^n\), the nth power of \(\mathbf{C}\). We determine conditions under which a given companion matrix \(\mathbf{C}\) is a primitive matrix. A systematic method for obtaining the limit values of a companion sequence is provided. A new class of primitive companion matrices is introduced for which the limit values of the related companion sequences are connected with the Golden ratio \(\uptau =\frac{1+\sqrt{5}}{2}\). In fact, a new generalization of the well-known \(\mathbf{Q}\)-matrix and the ordinary Fibonacci numbers are presented in this paper. This generalized form of the Fibonacci numbers will be referred to as the Golden-Fibonacci sequence. We show that the limit values of a Golden-Fibonacci sequence are powers of the Golden ratio. We apply primitive companion matrices as encoder matrices and introduce a type of error-correcting codes to be called companion coding. We show that the error-correcting relations of companion coding are connected with the limit values of the related companion sequence.

Keywords

Golden ratio Golden-Fibonacci sequence Companion matrix 

Mathematics Subject Classification

15B 11C 40B 94B 

Notes

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Copyright information

© Korean Society for Computational and Applied Mathematics 2019

Authors and Affiliations

  1. 1.Department of MathematicsMalek Ashtar University of TechnologyIsfahanIran
  2. 2.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran
  3. 3.Department of Electrical and Computer EngineeringUniversity of VictoriaVictoriaCanada

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