Equivalence of Butson-type Hadamard matrices

Two matrices H₁ and H₂ with entries from a multiplicative group G are said to be monomially equivalent, denoted by H₁≅ H₂, if one of the matrices can be obtained from the other via a sequence of row and column permutations and, respectively, left- and right-multiplication of rows and columns with elements from G. One may further define matrices to be Hadamard equivalent if H₁≅ ϕ(H₂) for some ϕ∈ Aut (G). For many classes of Hadamard and related matrices, it is straightforward to show that these are closed under Hadamard equivalence. It is here shown that also the set of Butson-type Hadamard matrices is closed under Hadamard equivalence.