A Generalization of Hadamard Matrices

Let S⊆C⁎=C\{0} and A∈Mn(C). The matrix A is called an S-GHMn if A∈Mn(S) and AA⁎=Diag(λ1,…,λn), for some positive numbers λi,i=1,…,n. In this paper we provide some necessary conditions on n for the existence of an S-GHMn over a finite set S. We conjecture that for every positive integer n, there exists a {±1,±2,±3}-GHMn.