Let ${x_1,x_2,...x_n}$ be positive numbers. Consider the matrix $C$ whose $(i,j)$-th entry is
$$\min\left\{\frac{x_i}{x_j},\frac{x_j}{x_i}\right\}$$
Show that $C$ is non-negative definite (or positive semidefinite, meaning $z^t C z\geq 0$ for all $z\in \mathbb{R}^n$).When is $C$ positive definite?