Some Remarks on Fibonacci Matrices

In [1], Dazheng studies Fibonacci matrices, namely matrices M such that every entry of every positive power of M is either 0 or plus or minus a Fibonacci number. He gives 40 such four-byfour matrices. In the following, we give an interpretation of these matrices, from which we give simpler proofs of several of his theorems. We also determine all two-by-two Fibonacci matrices. Let £ = e be a primitive fifth root of unity. Then £" is a root of the irreducible polynomial X + X + X + X + l, so the field Q(Q is a vector space of dimension 4 over Q with basis B = {1, Cj, ^, C} • The ring of algebraic integers in Q(£) is Z[£]. The units of this ring are of the fo rm( -£T^ , 0<m<9, TIGZ, where ^ = (1 + V5)/2 = H^ 2 +C 2 ) If a G Q ( ^ ) , then multiplication by a gives a linear transformation of Q(£), regarded as a vector space over Q, and hence a matrix M{a) with respect to the basis B. For example, let a = 0 = -(C+<;). Then