Prove normal matrices are RPN (Range perpendicular to Null space) Matrices?

Question

For a normal matrix $A$, $A^* A = A\ A^*$ where $A^*$ is the conjugate transpose of $A$
Prove that for normal matrices $$C(A) \perp N(A)$$

Answer 1

I will use the language of linear transformations. We have $\|A^{*}x\|^{2}=\langle A^{*}x, A^{*}x\rangle =\langle AA^{*}x, x\rangle =\langle A^{*}Ax, x\rangle ==\langle Ax, Ax\rangle =\|Ax\|^{2}$ which shows that for any $x$ in the null space of $A$ we also have $A^{*}x=0$. Now let $x$ be in the null space of $A$ and consider any vector $Ay$ in the range of $A$. We have $\langle Ay, x\rangle =\langle y, A^{*}x\rangle =\langle y, 0\rangle =0$.