Let $X\in\mathbb{R}^{n\times n}$ with $n\in\mathbb{N}^{+}$ be a random matrix with entries $x_{ij}$ continuous random variables for any $i,j\in\{1,...,n\}$ such that $i\neq j$, e.g., $x_{ij}\sim\mathcal{U}(0,1)$ uniformly distributed. The rowsum of $X$ is $0$ such that $$ x_{ii}=-\sum_{j=1,j\neq i}^{n}x_{ij} $$ How to prove that $X$ is diagonizable?
Intuitively, the probability that $2$ eigenvalues are equal for a (continuous) random matrix $X$ implying diagonalization is $0$ (almost surely); however, I cannot find a (concise) method to prove this or find out if this statement is true.