Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?

Question

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
$$C=\mathbf{F}\operatorname{Diag}(1,w,\cdots,w^{n-1})\mathbf{F}^{-1}$$ where $\mathbf{F}$ is just the discrete Fourier matrix and $w=\exp(\frac{2\pi i}{n})$

Let $U=(u_{ij})$ be a upper triangular matrix. We say it is supper upper triangular if $u_{i, i+1}=0$.

Let $U$ be a supper upper triangular whose entries are either 0 or 1. Is $C+U$ diagonalizable?

Answer 1

Here's a counterexample: For $U$ take $\left(\begin{smallmatrix} 0& 0 &0& 0& 1& 1& 1& 1\\ 0& 0 &0 &1& 0& 1& 1& 1\\ 0& 0 &0& 0& 1& 0& 0& 1\\ 0& 0 &0& 0& 0& 0& 0& 1\\ 0& 0 &0& 0& 0& 0& 1& 0\\ 0& 0 &0& 0& 0& 0& 0& 0\\ 0& 0 &0& 0& 0& 0& 0& 0\\ 0& 0 &0& 0& 0& 0& 0& 0 \end{smallmatrix}\right)$.

Then the rational canonical form of $C+U$ is $\left(\begin{smallmatrix} -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 &-1& 0 & 0 & 0 & 0 & 0 & 0\\ 0 &0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 &0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 &0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 &0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 &0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 &0 & 1 & 1 & -1 & 4& -2 & 2 \end{smallmatrix}\right)$