Eigenvalues of Circulant Matrices and a Conjecture of Ryser

Reinhardt Euler ; Luis H. Gallardo ; Olivier Rahavandrainy

Kragujevac Journal of Mathematics, Tome 45 (2021) no. 5, p. 751

Résumé

We prove that there is no circulant Hadamard matrix $H$ with first row $[h_{1},\ldots,h_{n}]$ of order $n>4$, under some linear conditions on the $h_{i}$'s. All these conditions hold in the known case $n=4,$ so that our results can be thought as characterizations of properties that only hold when $n=4.$ Our first conditions imply that some eigenvalue $\lambda$ of $H$ is a sum of $\sqrt{n}$ terms $h_{j}\omega^{j}$, where $\omega$ is a primitive $n$-th root of $1$. The same conclusion holds also if some complex arithmetic means associated to $\lambda$ are algebraic integers (second conditions). Moreover, our third conditions, related to the recent notion of \emph{robust} Hadamard matrices, implies also the nonexistence of these circulant Hadamard matrices. If some of the conditions fail, it appears (to us) very difficult to be able to prove the result.

Détail

Classification : 11B30 15B34, 11R18
Keywords: circulant matrices, Hadamard matrices, eigenvalues, unit circle, cyclotomic fields