A Generalization of Hadamard Matrices

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