My question is different than this question because the latter asks whether it is possible, but I am asking under what general condition(s) it is possible.
Question: Under what condition is the sum of two (symmetric) positive semidefinite matrices positive definite? Let $A$ and $B$ be two $n \times n$ positive semi-definite matrices, i.e., $x^{\top}Ax \geq 0$ and $y^{\top}By \geq 0$ for all $\mathbb{R}^n \ni x\neq 0 \neq y \in \mathbb{R}^n$. What condition on $A$ and $B$ guarantees $x^{\top}(A+B)x > 0$ for all $0 \neq x \in \mathbb{R}^n$?
My thoughts
If there is $0 \neq x \in \mathbb{R}^n$ such that $x^{\top}(A+B)x > 0$, this means that $x \in \text{Null}(A)$ and $x \in \text{Null}(B)$, i.e., or $\text{Null}(A) \cap \text{Null}(B) \neq \emptyset$. However, this condition is not practical. For example, let $A=vv^{\top}$ and $B=ww^{\top}$ for $0 \neq v \in \mathbb{R}^n$ and $0 \neq w \in \mathbb{R}^n$. How can I find a condition on $v, w$? I know in this case, the range of $A$ and $B$ is span{$v$} and span{$w$} but we want the entire space be covered.