on $\mathbb{C}^{2}$ we define the inner product $ < x,y>=x \begin{pmatrix}
1 &\imath \\ 
 -\imath & 2
\end{pmatrix} y^{\ast }$

i want to show that $ < x,y>$ $ \geq 0$

I consider $x$ to be $x=(x\_1,x\_2)$ and i calculate $ < x,x>=\left | x\_1 \right |^{2}+2\left | x\_2 \right |^{2} +i(\bar{x\_2}x\_1-x\_2\bar{x\_1})$ but that's a complex number and it doesn't make sense to have inequalities on complex numbers. Then I thought to just take the norm, but the norm is defined to be $\geq 0$ . Can someone make it more clear for me ?