I am reviewing an old Linear Algebra textbook, and am going through a chapter about going from a Quadratic Form to a Conic Section.
For example, starting with the following Quadratic Form:
Then find a symmetric Matrix, A, such that
Then find an orthogonal matrix, Q, such that , where D is a diagonal matrix.
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: from a Conic to a 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?