Prove that $\lambda_\max (yy^*)=\|y\|^2_2$

Question

Problem: Let $y\in \mathbb{C}^n$, prove that $\lambda_{\max}(yy^*)=\|y\|^2_2$ where $y^*=\overline{y^T}, \|y\|^2_2=y^*y.$

I know that $\|y\|_2$ is an e-value of $yy^*$ since $(yy^*)y=\|y\|_2y$, but I could not go further. Help me, Thanks!