Relation between singular value, eigen value and 2-norm

Question

I am confused. We have that, $ \sigma_{max} \ge \rho(A) = \|A\|_2$.

where $ \rho(A) = |\lambda_{max}|$ and $\sigma$ is the singular value of A. And, $\sigma^2 = \lambda(A^*A)$.

But, $\rho(A) \ne \sqrt{\rho(A^*A)}$.