Suppose one is given a set of positive real numbers $\{ a_i >0 \}_{i=1}^S$ and set of vectors $\{ x_i \}_{i=1}^S$. Now consider the matrix, $M = \sum_{i=1}^S a_i x_i x_i^\top$.
What are some of the easy to state conditions so that $M$ is positive definite?
Suppose that the matrix $M_x = \sum_{i=1}^S x_i x_i^\top$ is given to be positive definite.
Then what can be a condition on the numbers $\{ a_i >0 \}_{i=1}^S$ so that $M$ is positive definite?