The text of the exercise is the following;
Consider $q(X)=2x_1x_3 + 2x_2x_3 − 2x_1x_2 − x_1^2-x_3^2$ a quadratic form first on $\mathbb R^3$ and after on $\mathbb C^3$
find the matrix $G = [g]_E$, where $E$ is the standard basis and $g$ is the symmetric form
find a $g$-orthogonal basis $B$ and then write the matrix $[g]_B$ with respect to basis $B$.
My professor said this is a quadratic symmetric form , so in $\mathbb R^3$ to write the matrix is quite simple $G = [g]_E$= $\begin{pmatrix}-1&-1&1\\-1&0&1\\1&1&-1\end{pmatrix}$
In order to find a g-orthogonal basis I have first to find a basis of annihilator $b_1=\begin{pmatrix}1&0&1\end{pmatrix}$, then a complementary basis $b_2=(\begin{pmatrix}0&0&1\end{pmatrix}$ $\begin{pmatrix}0&1&0\end{pmatrix})$, use Gram-Schmidt on the third vector in this case and then I have obtained $[g]_B$=$\begin{pmatrix}1&0&0\\0&0&1\\1&1&1\end{pmatrix}$
Now I am asking what is the difference in the complex case? My professor said that in $\mathbb C^3$ I have to do the same steps, annihilator etc... but I will have different matrices, but I don't even know how to do the first point.