When working with a change of coordinates using
$x^TAx=k$
how and when do we deal with translations? I'm comfortable with setting up the formula $x^TAx$ where A is the matrix whose diagonal entries are the coefficients of (say in $R^2$) x and y and whose other entries correspond to half of the $xy$ term in said equation. Then finding eigenvectors that translate our coordinate system and sketching our graph from there. What I'm struggling with is say we have:
$x^2 - 2xy + y^2 + 6x - 4 = 0$
I know that if I were to say, complete the square I would have something that looks like:
$(x+3)^2 - 2xy + y^2 = 13$
Do I then just move forward and still build my matrix A as \begin{align} A = \begin{pmatrix} 1 & -1 \\ -1 & 1 \\ \end{pmatrix} \end{align}
Or do I need to account for the shift when building this matrix? Again, I'm comfortable with the math up until dealing with the translation. Do I complete the process of sketching the graph and then account for my translation or does the translation play a role throughout the process?