Spectra of Hadamard matrices

A Butson Hadamard matrix $H$ has entries in the kth roots of unity, and satisfies the matrix equation $HH^{\ast} = nI_{n}$. We write $\mathrm{BH}(n, k)$ for the set of such matrices. A complete morphism of Butson matrices is a map $\mathrm{BH}(n, k) \rightarrow \mathrm{BH}(m, \ell)$. In this paper, we develop a technique for controlling the spectra of certain Hadamard matrices. For each integer $t$, we construct a real Hadamard matrix $H_{t}$ of order $n_{t} = 2^{2^{t-1}-1}$ such that the minimal polynomial of $\frac{1}{\sqrt{n_{t}}}H_{t}$ is the cyclotomic polynomial $\Phi_{2^{t+1}}(x)$. Such matrices yield new examples of complete morphisms $$\mathrm{BH}(n, 2^{t}) \rightarrow \mathrm{BH}(2^{2^{t-1}-1}n, 2)\,,$$ for each $t \geq 2$, generalising a well-known result of Turyn.