I have to convert some quadratic forms into real and complex canonical forms.
One of these forms is as such:
$q_1\begin{pmatrix} x \\ y \\ z \end{pmatrix} = x^2 +3y^2 +z^2 +2xy−2xz−2yz$
From the matrix $\begin{Bmatrix} 1 & 1 & -1 \\1 & 3 & -1 \\-1 & -1 & 1 \end{Bmatrix}$
I got $\begin{Bmatrix} 1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & -2 \end{Bmatrix}$ using column and row operations but I am not sure that this is correct? Is it the real form? If so, how do I get the complex form?
I also have to compare this equation to others and say which are equivalent over $\mathbb{R}$ and $\mathbb{C}$. How would I go about this?
Edit: I worked out the both the real and complex form for this one to have $1,1,0$ on the diagonal. I have two other quadratic forms $q_2$ and $q_3$ for which I worked out the diagonals for the real form as $1,1,-1$ and $-1,-1,-1$ respectively. Can I just change the $-1$s to $+1$s to make them complex?
I am also asked which of $q_1,q_2,q_3$ are equivalent over $\mathbb{R}$ and $\mathbb{C}$. If the real form is unique then how can they be equivalent? Am I right in thinking that none of the real forms are equivalent but the complex forms of $q_2$ and $q_3$ are?