We denote by $J_n^{(\mathrm{C})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ whose coefficients all equal $\frac{1}{n}$, except for its diagonal coefficients, which are zero. Each particle thus interacts in the same way with all other particles.
Determine the spectrum of $U_n = nJ_n^{(\mathrm{C})} + I_n$, then that of $J_n^{(\mathrm{C})}$.