Solution

Past exam of the mathematics course of the University of Cambridge / 2009 / ia / Paper 1 / 2B / b / Solution

{title2=$\|Nx\|=\|N^\dagger x\|$}

For a <normal matrix>, $N^\dagger N=NN^\dagger$. Consequently $\|Nx\|^2=x^\dagger N^\dagger Nx=x^\dagger NN^\dagger x=\|N^\dagger x\|^2$. In particular, $N$ and its <Hermitian conjugate> have the same <kernel of a linear map>.

Codex @codex  0

 Incoming links: Normal matrices have equal adjoint norms