A real $n\times n$ matrix $X$ is called positive semidefinite (resp. positive definite) if it satisfies $$ X = X^{\top}, \quad v^{\top} X v \geq (\text{resp.} >) \, 0, \qquad \forall \, v \in \mathbb{R}^{n}.$$
(10 points) Prove that all eigenvalues of $X$ are $\geq 0$ for positive semidefinite case, and $>0$ for positive definite case.
(10 points) Let $\mathbb{S}_{+}^{n}$ denote the set of all $n\times n$ positive semidefinite matrices. Prove that $\mathbb{S}_{+}^{n}$ is a convex cone.
(Hint: Choose some arbitrary positive integer $m$. Then from the definition of positive semidefiniteness, prove that for any $X_{1}, X_{2}, ..., X_{m} \in \mathbb{S}_{+}^{n}$, and $\alpha_{1}, \alpha_{2}, ..., \alpha_{m} \geq 0$, the matrix $\sum_{i=1}^{m}\alpha_{i}X_{i} \in \mathbb{S}_{+}^{n}$.)
(10 points) Let $A, B \in \mathbb{S}_{+}^{n}$. True or false: $AB \in \mathbb{S}_{+}^{n}$? If your answer is "true", then give a proof. If your answer is "false", then give a counterexample.
(10 points) Let $A, B \in \mathbb{S}_{+}^{n}$. The Hadamard product $A\odot B$ is defined as the element-wise product: $(A\odot B)_{ij} := a_{ij}b_{ij}$. True or false: $A\odot B \in \mathbb{S}_{+}^{n}$? If your answer is "true", then give a proof. If your answer is "false", then give a counterexample.