As a follow-up to this question I asked, I wondered what would happen if I imposed the weaker condition of having positive eigenvalues, rather than being positive definite.
How do I construct an example of two matrices $A$ and $B$ such that:
1) $A$ and $B$ have strictly positive eigenvalues.
2) $A + B$ has strictly negative eigenvalues (is this even possible?).
3) $AB$ has strictly negative eigenvalues.
Generally, I'm unsure how to begin going about constructing an example of a matrix that satisfies these properties.