By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that $$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis in $\mathbb{R}^n$. It is well-known that $C$ is diagonalized by the discrete Fourier matrix.

Let $U$ be a strictly upper triangular matrix. Is $C+U$ diagonalizable?