I am working on a problem in which I must classify the surface described by the following equation
$$x^2 + y^2 - z^2 + 2xy - 2xz - 2yz - y = 0.$$
I have looked at this Stack Exchange discussion (on the very same question).
Now say we let $A$ be the matrix associated with the quadratic form obtained by ignoring the term $-y$ in the equation above.
The above discussion states that, once the eigenvectors and eigenvalues of $A$ have been found, all that remains is to put 'it in the central form'.
Firstly, what does 'it' refer to (the original equation?)?
Secondly, could I have a hint as how to manipulate 'it' now that the eigenvectors and eigenvalues have been found? I am not sure how to go about this yet.