Write the quadratic form $\displaystyle x_1x_2 +x_2x_3 +x_3x_1$ in canonical form over $\displaystyle \mathbb{C}$.

I started off by finding a symmetric matrix A for the quadratic form $\displaystyle Q(\vec{x},\vec{x})=\vec{x}^TA\vec{x}$, which is

$\displaystyle \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0\end{pmatrix}$ with eigenvalues of $\displaystyle \lambda=-.5, -.5, 1$ and eigenvectors $\displaystyle (-1,1,0), (-1,0,1), (1, 1, 1)$ respectively.

I don't see how it's now possible to rewrite the expression as $\displaystyle Q(\vec{x},\vec{x})=\lambda_1x_1^2+\lambda_2x_2^2+\ lambda_3x_3^2$. And where do complex numbers have anything to do with the problem?