Canonical form of a Quadratic form. I've been given the following quadratic form to find the canonical form of:
$$
Q(\bf{z})= z_1z_2 + 2z_2z_3 − 3z_3z_4
$$
through the method of forming perfect squares.
The method I've been taugh/show is to look at terms consisting of a specific variable, say $z_1$ and form the perfect square. We then set the canonical basis as the inside of the square (hopefully that makes sense...)
Until we get the quadratic form to look like:
$$
Q(z)=\alpha_1(\eta^1)^2+\alpha_2(\eta^2)^2+\alpha_3(\eta^3)^2+\alpha_4(\eta^4)^2
$$
where the alphas are the canonical coefficients.
Now I am stuck with the above problem as there is no square term. Typically my approach in these problems has been to start with a term that has a square term, and go from there. In this case, whenever I get to the final $\eta$ to find, I am left with two square terms, say $z_3$ and $z_4$
Is there something im missing?

Gantmacher's version of Lagrange's method leads to
$$ \left( \frac{1}{2} x_1 +  \frac{1}{2} x_2 + x_3 \right)^2 - \left( -\frac{1}{2} x_1 +  \frac{1}{2} x_2 - x_3 \right)^2  +  \left( \frac{1}{2} x_2 +  \frac{1}{2} x_3   -\frac{3}{2} x_4 \right)^2 -  \left( -\frac{1}{2} x_2 +  \frac{1}{2} x_3   +\frac{3}{2} x_4 \right)^2$$




