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 vectors in $\mathbb{R}^n$. It is well-known that
$$C=F\operatorname{Diag}(1,w,\cdots,w^{n-1})F^{-1}$$ where $F$ is just the discrete Fourier matrix and $w=\exp(\frac{2\pi i}{n})$

Let $U=(u_{ij})$ be a strictly upper triangular matrix with $u_{ij}\in \{0,1\}$ . Is $C+U$ diagonalizable?