An application of matricial Fibonacci identities to the computation of spectral norms

Abstract

Among the most intensively studied integer sequences are the Fibonacci and Lucas sequences. Both are instances of second order recurrences [8], both satisfying sk−2+sk−1 = sk for all integers k, but where the fibonacci sequence (fi) begins with f0 = 0 and f1 = 1, the Lucas sequence (li) has l0 = 2 and l1 = 1. Several authors have recently been interested in the singular values of Toeplitz, circulant, and Hankel matrices that are obtained from the Fibonacci and Lucas sequences (see [1], [2], [3], [12], [13], and [14]), where the authors obtain bounds for the “spectral norms”, i.e. the largest singular value. In [1] a formula is given for the exact value of the spectral norms of the Lucas and Fibonacci Hankel matrices. In this paper, we present the exact value for the spectral norms of Toeplitz matrices involving Fibonacci and Lucas numbers. All matrices and vector spaces will be considered as ones over the complex numbers. If A is a complex matrix, we let A∗ denote the adjoint of A. The singular values of a matrix A = (aij) are defined to be the nonzero eigenvalues of |A| ≡ (A∗A) 1 2 , and they are traditionally enumerated in descending order, s1 ≥ s2 ≥ . . . ≥ sk > 0,