I am reviewing an old Linear Algebra textbook, and am going through a chapter about goingfroma Quadratic Formtoa Conic Section.

For example, starting with the following Quadratic Form:

Let

Then find a symmetric Matrix, A, such that

Then find an orthogonal matrix, Q, such that , where D is a diagonal matrix.

Also, let

At this point, we have enough information to determine if the original Quadratic Form represented a parabola, ellipse, or hyperbola.

However, I would like to go in the reverse direction:froma Conictoa Quadratic Form.

For example, say I have an ellipse of eccentricity,e.

How can I work backwards to create the Q, D, and A matrices, compute their eigenvalues and eigenvectors, etc.

Can anybody suggest an easy-to-understand reference for working through these steps?