Spectra of Hadamard matrices

Abstract

A Butson Hadamard matrix $H$ has entries in the kth roots of unity, and satisfies the matrix equation $H{H}^{\ast }=n{I}_n$. We write $\mathrm{BH}(n,k)$ for the set of such matrices. A complete morphism of Butson matrices is a map $\mathrm{BH}(n,k)\to \mathrm{BH}(m,\ell )$. 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 $t\ge 2$, generalising a well-known result of Turyn.

12 pages