Let $X$ be a $d$-dimensional random vector, and $\beta$ a vector. When is the following matrix positive definite?
$$\mathbb{E}\left[ \frac{e^{X^T \beta}}{1 + e^{X^T \beta}} XX^T\right]$$
My notes says it is positive definite if $\mathbb{E}\left[XX^T\right]$ is non-singular, but I don't see why is that? My notes could be wrong though...
Take care that exponential term is only a positive value.but the whole can never be a positive definite matrix. rank of multiplication of two matrix $A$ , $B$ follows this relation: $$rank(AB) \le min (rank(A),rank(B))$$ So rank of the term $XX^T$ can not be more than 1 and it has $d-1$ zero eigenvalue. It is only a positive semidefinite matrix and alway it is singular.