I'm doing exercises on the reductions of conics to canonic forms using eigenvalues. I'm trying to understand what does actually change when I put the two eigenvalues that I find in a different order
For example:
$\mathscr{C} : x^2-xy+\frac{1}{4}y^2-2x+6y+6=0$
is a parabola and the matrix $A$ associated to it has eigenvalues $0$ and $\frac{5}{4}$
To reduce the conic I have to choose the order of the eigenvalues to put in the reduced form and the first eigenvalue is the coefficient of the $x^2$ term so in one case the parabola will have only the $x^2$ and in the other only the $y^2$
Does this make any difference? Is there a particolar order that must be followed? Is the conic just rotated in two different ways?
Thanks a lot for your help
Indeed, the quadratic terms are $(2x-y)^2/4$, which you can rewrite as $u^2/4$. There is no choice between $x$ or $y$, this is a new coordinate.
The rotation is unambiguously defined by $u=2x-y$.