Equivalent Forms

Question

I want to show that $2xy+z^2$ is equivalent to $x^2-y^2+z^2$ as quadratic forms over $\mathbb{Z}$. This means I want to use an integral linear change of variables to get from one to the other. I'm struggling with this. My approach is diagonalizing the associated matrix (but this gives me noninteger coefficients when I do the change of variables). Is this an obvious change of variables to make?

Answer 1

$2xy+z^2 = (x-z)^2+x(2y+2z-x) = (x-z)^2 - (y+z-x)^2+(y+z)^2$