Normal Matrix: Norm vs. Spectral Radius

Question

Define the following matrix norm: $$ ||A||=\frac{\max|\langle Ax,x\rangle|}{||x||_2^2} $$ The spectral radius of $A$ is: $$ \rho(A)=\max_i\{|\lambda_i|:Ax_i=\lambda_ix_i; x_i \neq 0\} $$ Can you provide me some hints to show that: $$ ||A||=\rho(A) \text{ when } AA^T=A^TA? $$

Answer 1

Since $A$ is normal, the spectral theorem applies. Now what happens if you write an arbitrary vector in terms of the eigenvector basis guaranteed by the spectral theorem?