We know that the below inequality holds if A is positive definite
${\lambda _{\min }}\left( {{A}} \right){\left\| x \right\|^2} \le {x^T}{A}x$ or equivalently $\alpha{\left\| x \right\|^2} \le {x^T}{A}x$ where $\alpha>0$
Is there any other condition that the above inequality can still hold?