Equivalence of Butson-type Hadamard matrices

Abstract

AbstractTwo matrices $$H_1$$ H 1 and $$H_2$$ H 2 with entries from a multiplicative group G are said to be monomially equivalent, denoted by $$H_1\cong H_2$$ H 1 ≅ H 2 , 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 $$H_1 \cong \phi (H_2)$$ H 1 ≅ ϕ ( H 2 ) for some $$\phi \in \mathrm {Aut}(G)$$ 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.