Relation between norms and inner product

Question

I'm trying to show that for $\mathbf{x} \in \mathbb{C}^N$ and $\mathbf{A} \in \mathbb{C}^{m\times N}$ that $$ \|\mathbf{Ax}\|_2^2-\|\mathbf{x}\|_2^2=\langle(\mathbf{A}^\ast \mathbf{A}-\mathbf{I})\mathbf{x},\mathbf{x}\rangle.$$

I understand why this is true for the real setting, but I'm stuck with the complex setting.

Answer 1

$\|Ax\|_2^2 = \langle Ax, Ax \rangle = \langle A^* A x, x\rangle$

$\|x\|_2^2 = \langle x,x\rangle$

Therefore, $\|Ax\|_2^2 - \|x\|_2^2 = \langle A^* A x, x\rangle - \langle x,x\rangle$ which leads to the desired expression.