In $\mathbb{C}^2$, I want to show that $\langle x,y \rangle =xAy^*$ is an inner product space. I almost done with other properties of inner product but left only one property: $\langle x,x \rangle >0$ if $x \neq 0$.
To prove this,
$$\begin{align*}
\langle x,x \rangle&=(x_1, x_2)A(x_1, x_2)^*\\
&=\|x_1 \|^2-ix_2\overline{x_1}+ix_1\overline{x_2}+\|x_2 \|^2\\
&=\|x_1 \|^2 +2\operatorname{Re}(ix_1\overline{x_2})+\|x_2 \|^2
\end{align*}$$
where $\operatorname{Re}(z)$ is the real part of the complex number $z$.
I can't follow the last equality. Why does $\operatorname{Re}(z)$ appear here?
For any complex number $w$, we know that $w + \bar w = 2 \Re(w)$.
Apply this with $w = ix_1\bar x_2$.