Companion matrices and Golden-Fibonacci sequences

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.