We denote by $J_n^{(\mathrm{s})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by
$$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad J_n^{(\mathrm{S})}(i,j) = \frac{2}{\sqrt{2n+1}} \sin\left(\frac{2\pi ij}{2n+1}\right).$$
Deduce that $J_n^{(\mathrm{s})}$ is a symmetric orthogonal matrix.