"Positive definite" matrix for component-wise positive vector

Question

Is there a condition such that a symmetric matrix is "positive definite" for any component-wise positive vector?

There is a matrix $A$, and $A \in S^n$, $S$ denotes the real symmetric matrix set. What's the condition for $x^T A x > 0$, and $x \in R^n$, $|x| \neq 0$, and $x \succeq 0$ (component-wise positive).

Thank you.

Answer 1

Look up copositive matrices. A matrix $A$ is copositive if $x^{T}Ax \geq 0$ for all $x \geq 0$.