Reduced and canonical equations of the conics
Find the canonical equation of the conic defined by the following equation $x^2+y^2+2x+3=0$.
To begin with, notice that there is no term $xy$, so the first reduction is not necessary because the main matrix $A'$ is already diagonal.
Completing the squares for $x$, we see that the equation becomes $$(x+1)^2+y^2+2=0$$
Doing the change of variable $x' = x+1, \ y' = y$ we are left with the equation $$x'^2+y'^2+2=0$$ Notice that the canonical equation is that of an imaginary ellipse.
The canonical equation is $x'^2+y'^2+2=0$ and therefore this is an imaginary ellipse.