$A - aI$ positive definite for some $a > 0$

Question

If $A$ is positive definite and $I$ is the identity, does there exist an $a > 0$ such that $A - aI$ is positive semi definite.

The context for this is I am wanting to find a $c > 0$ such that $cx^Tx \leq x^TAx$ for all $x \in \mathbb{R}^n$ if A is positive definite.

Answer 1

Yes: a symmetric matrix $A$ is positive definite iff all the eigenvalues are positive, so if $a$ is anything less than the smallest eigenvalue of $A$, then $A-aI$ is positive definite.