The linear algebra of the k-Fibonacci matrix

For a positive integer k 2, the k-Fibonacci sequence {g(k)n} is defined as: g(k)1= =g(k)k−2=0, g(k)k−1=g(k)k=1 and for n>k 2, g(k)n=g(k)n−1+g(k)n−2+ +g(k)n−k. The n×n k-Fibonacci matrix is defined as: for fixed k 2, where gn=g(k)n+k−2. Also, the n by n k-symmetric Fibonacci matrix is defined as where q(k)ij=0 for j 0. If k=2, then is the Fibonacci matrix and is the symmetric Fibonacci matrix. The properties of the Fibonacci matrix and the symmetric Fibonacci matrix are well-known. In this paper, we discuss the linear algebra of the k-Fibonacci matrix and the symmetric k-Fibonacci matrix.