Triangular truncation of k-Fibonacci and k-Lucas circulant matrices

摘要

We prove a general theorem that gives tight bounds on the spectral norms of triangularly truncated k-Fibonacci and k-Lucas circulant matrices. The bounds are good enough to enable the calculation of the limit ‖C‖/‖τ(C)‖ as the dimension n approaches infinity, where τ(C) denotes the triangular truncation of C,and C is any n x n circulant matrix built using a sequence (s_i) satisfying S_i = kS_(i-1) + S_(i-2). In particular, we have that this limit is equal to the golden ratio, if C is built using either the ordinary Fibonacci or Lucas sequence.