# A Real eigenvector of Circulant matrices and a conjecture of Ryser

## TL;DR

This paper proves the non-existence of circulant Hadamard matrices of order greater than 4 under specific conditions related to scalar products of associated circulant matrices, advancing understanding in matrix theory.

## Contribution

It introduces a new condition involving scalar products of related circulant matrices to prove the non-existence of certain Hadamard matrices.

## Key findings

- No circulant Hadamard matrix of order > 4 exists under the given condition.
- The proof relies on properties of eigenvectors of circulant matrices.
- The result narrows the search for circulant Hadamard matrices.

## Abstract

We prove that there is no circulant Hadamard matrix $H$ with first row $[h_{1},\ldots,h_{n}]$ of order $n>4$, under a condition about a sum of scalar products of rows of two other circulant matrices of size $n/2$ associated to $H.$