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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?

  • 2
    Non-negative definite? Is that another way of saying "positive semidefinite"?2012-09-17
  • 0
    Where is the problem from?2012-09-17

3 Answers 3