relationship between the $2$-norm of a symmetric matrix and its maximum eigenvalue. If I have an $N\times N$ symmetric matrix $Q$, what is the relationship between $\|Q\|$ and its maximum eigenvalue, where $\|\|$ is the $2$nd norm?

The general answer to this is this $\|Q\| = \max \{|\lambda_{\max}(Q)|, |\lambda_{\min}(Q)|\}$. To see why this is true, consider $Q = I_{N}$ for the first case and $Q = -I_{N}$ for the second case. 
