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Abstract

A new proof is given of Newman and Taussky's result: if A is a unimodular integral n x n matrix such that A′A is a circulant, then A = QC where Q is a generalized permutation matrix and C is a circulant. A similar result is proved for unimodular integral skew circulants. Certain additional new results are obtained, the most interesting of which are: 1) Given any nonsingular group matrix A there exist unique real group matrices U and H such that U is orthogonal and H is positive definite and A = UH; 2) If A is any unimodular integral circulant, then integers k and s exist such that A′ = P(k)A and P(s)A is symmetric, where P is the companion matrix of the polynomial xⁿ-1. Finally, all the n x n positive definite integral and unimodular skew circulants are determined for values of n ≤ 6: they are shown to be trivial for n = 1,2,3 and are explicitly described for n = 4,5,6.