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

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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×n circulant matrix built using a sequence (si) satisfyingsi=ksi−1+si−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.