Trace, Symmetry and Orthogonality

Does there exist a circulant conference matrix of order > 2? When is there a symmetric Hadamard matrix with constant diagonal? How many pairwise disjoint, amicable weighing matrices of order n can there be? These are questions concerning which the trace function gives a great deal of insight. We offer easy proofs of the known solutions to the first two, the first being new, and develop new results regarding the latter question. It is shown that there are 2 t disjoint amicable weighing matrices of order 2 tp, where p is odd, and that this is an upper bound for t ≤ 1. An even stronger bound is obtained for certain cases.