 Normal matrices have equal adjoint norms

For a normal matrix, $N^{†}N=N{N}^{†}$. Consequently $\Vert Nx{\Vert }^2=x^{†}{N}^{†}Nx=x^{†}N{N}^{†}x=\Vert N^{†}x{\Vert }^2$. In particular, $N$ and its Hermitian conjugate have the same kernel of a linear map.