Let $\alpha,\beta\in \mathbb{R}^n$. Prove that if $\alpha' \beta \gt 0$, then there exists a positive definite matrix $A$ such that $A\alpha=\beta$.
My attempt:take $A= \frac{\beta\alpha'}{\alpha'\alpha}$, however I can only prove that $A$ is non-negative definite. How can I move on?