The linear algebra of the k-Fibonacci matrix

Abstract

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 F(k) n=[f(k) ij] n is defined as: for fixed k⩾2, f(k) ij= g i−j+1 i−j+1⩾0, 0 i−j+1<0, where g n = g( k) n+ k−2 . Also, the n by n k-symmetric Fibonacci matrix Q(k) n=[q(k) ij] n is defined as q(k) ij=q(k) ji= ∑ l=1 kq(k) i,j−l i+1⩽j, ∑ l=1 kq(k) i,i−l+g 1 i=j, where q( k) ij =0 for j⩽0. If k=2, then F(2) n is the Fibonacci matrix and Q(2) n 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.