By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that
$$ C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right] $$
where $e_1,\dots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. Let $U=(u_{ij})$ be a upper triangular matrix. We say it is super upper triangular if $u_{i, i+1}=0$.
Here, we just focus on the super upper triangular matrices $U$ whose entries $u_{i,j}$ are either $0$ or $1$.
Q. Can we characterize super upper triangular matrices $U$ such that $C+U$ are diagonalizable?
Remarks:
In this MathOverflow post, an example of non-diagonalizable $C+U$ is given when the size $n=8$.
We (checked by MATLAB) found out the following result by experience: If $n\leq6$, all matrices $C+U$ are diagonalizable. When $n=7$ among all sums, there are just $518$ cases such that $C+U$ are not diagonalizable.
One may check that $C+U$ is diagonalizable if and only if the roots of the characteristic polynomial are all simple.
