Define $||A|| \equiv \max \{ ||Ax|| : ||x|| = 1 \}$. Let $A$ be hermitian. We wish to show that $||A||$ is equal to the largest eigenvalue. The proof idea is to consider the eigenvectors $v[i]$ with eigenvalue $\lambda[i]$ with largest eigenvalue $v^\star$ of eigenvalue $\lambda^*$ and claim that $||Av^\star|| = \lambda^*$ is maximal.