= Solution
A complex square matrix $A$ is a <normal matrix> if it commutes with its <adjoint matrix>[conjugate transpose]:
$$
AA^*=A^*A.
$$
By <unitary diagonalization of a normal matrix>, there are a <unitary matrix> $U$ and a diagonal matrix $\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_N)$ such that
$$
A=U\Lambda U^*.
$$
The <matrix 2-norm> is invariant under multiplication by unitary matrices, so
$$
\lVert A\rVert_2=\lVert\Lambda\rVert_2.
$$
For a diagonal matrix,
$$
\lVert\Lambda x\rVert_2^2
=\sum_j|\lambda_j|^2|x_j|^2
\leq\left(\max_j|\lambda_j|^2\right)\lVert x\rVert_2^2,
$$
with equality on a coordinate vector belonging to an eigenvalue of greatest modulus. Hence
$$
\boxed{\lVert A\rVert_2=\max_j|\lambda_j|=\rho(A)},
$$
the <matrix 2-norm of a normal matrix>.
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