If matrix $A-I$ is positive semidefinite, does $\lambda_{\inf} \geq 1$ hold

Question

If matrix $A-I$ is positive semidefinite, does the following hold?

$$\lambda_{\inf} \geq 1$$

where $\lambda_{\inf}$ is the infimum of the set of all eigenvalues of $A$. If so, why?

Thanks in advance.

Answer 1

HINT: Suppose you had an eigenvector $v$ with corresponding eigenvalue $\lambda$. What does $\langle (A-I)v,v\rangle\ge 0$ tell you?