Rotation matrix to construct canonical form of a conic
I've found C is a non-degenerate ellipses (computing the cubic and the quadratic invariant), and then I've studied the characteristic polynomial
The eigenvalue are , with associated eigenvectors . Thus I construct the rotation matrix R by putting in columns the normalized eigenvectors (taking care that ):
Then , and after some computations I find the canonical form
When you are told to work with conic sections equations, think about how planets travel around the sun and how elliptical routes get in focus. Parabolic mirrors, solar explanations, and so on. We have conic sections practice problems with answers that will help you learn easier. In the majority of cases, one should start with conic section equations because it is where one must start. Alternatively, take your conic section equation and compare things to the answers that we have below. See provided conic sections examples as these solutions will provide you with great starting points.