Unitary matrices associated with Butson-Hadamard matrices

Abstract

We give an example of a BH(5; 5) matrix M satisfying that the eigenvalues of the unitary matrix $\frac{1}{\sqrt{5}}M$ are all primitive 10-th roots of unity and such that $\frac{1}{5}{M}^3$ is not a BH(5; 5) matrix. This example gives a negative answer to a conjecture proposed by R. Egan and P. O. Cathain.