{"paper":{"title":"Spectra of Hadamard matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eric Swartz, Padraig O Cathain, Ronan Egan","submitted_at":"2018-07-11T16:38:17Z","abstract_excerpt":"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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.04238","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}