How do I find the canonical form of
$$q_1(x,y,z)= 4x^2 +4xz+2yz$$
Now I have put it in matrix form as:
$$\left( \begin{matrix} 4 & 0 & 2 \\ 0 & 0 & 1 \\ 2 & 1 & 0 \\ \end{matrix} \right) $$
So my question now is what do I do next? I know that the complex form canonical form has mixture of $1$ and $-1$ and the complex form only has 1 in it's diagonal. Do I just row reduce to get that form? Thanks.
$$q_1=4x^2+4xz+2yz=\left(\begin{matrix}x&y&z\end{matrix}\right) \left(\begin{matrix} 4 & 0 & 2 \\ 0 & 0 & 1 \\ 2 & 1 & 0 \\ \end{matrix}\right)\left(\begin{matrix}x\\y\\z\end{matrix}\right) $$
By Gauss method we have $$4x^2+4xz+2yz=(2x+z)^2-z^2+2yz=(2x+z)^2-(z-y)^2+y^2=\varphi_1^2(x,y,z)+\varphi_2^2(x,y,z)-\varphi_3^2(x,y,z)$$ so the signature is $(1,1,-1)$ and we know that the linear forms $\varphi_i$'s are linearly independant and if we denote by $\mathcal{B}=(e_1,e_2,e_3)$ the dual basis of the basis $(\varphi_1,\varphi_2,\varphi_3)$ then the matrix of $q_1$ in $\mathcal{B}$ is $$\left(\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \\ \end{matrix}\right)$$