On the matrix powers and exponential by the \(r\)-generalized Fibonacci sequences methods: The companion matrix case
An \(r\)-generalized Fibonacci number \((r\geq 2)\) is a sequence \((V_{n})_{n\geq 0}\) of real (or complex) numbers defined by \(V_{i}=\alpha _{i}\), \(i=0,\ldots ,r-1\) and \(V_{n+1}=\alpha _{0}V_{n}+\cdots +\alpha _{r-1}V_{n-r+1}\), \(n\geq r-1\) [see, for example, \textit{G. N. Philippou}, Proc. 2nd Int. Conf., San José/Cal., August 1986, 89-96 (1988; Zbl 0651.10008)]. Using some \(r\)-generalized Fibonacci sequences in the algebra of square matrices \(GL(n,{\mathbb C})\), some explicit formulas for the matrix powers and exponential are given, particulary for a companion matrix. In the case of the calculus of the power matrix, a result of \textit{W. Y. C. Chen} and \textit{J. D. Louck} [Linear Algebra Appl. 232, 261-278 (1996; Zbl 0838.15015)] is obtained and in the case of the calculus of the exponential matrix, a connection with the method of \textit{L. Verde-Star} [Stud. Appl. Math. 91, 153-177 (1994; Zbl 0810.47008)] is found.
scientific article; zbMATH DE number 1970629
scientific article