A Real eigenvector of Circulant matrices and a conjecture of Ryser

We prove that there is no circulant Hadamard matrix $H$ with first row $[h_1,\dots ,h_n]$ of order $n>4$, under a condition about a sum of scalar products of rows of two other circulant matrices of size $n\mathrm{/}2$ associated to $H\mathrm{.}$