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:

$\displaystyle F(x,y) = a x^{2} + b x y + c y^{2}$

Let $\displaystyle \vec{v} = \begin{bmatrix}x\\ y\end{bmatrix}$

Then find a symmetric Matrix, A, such that $\displaystyle F(x,y) = A \vec{v}\cdot \vec{v}$

Then find an orthogonal matrix, Q, such that $\displaystyle Q^{T}A Q = D$, where D is a diagonal matrix.

Also, let $\displaystyle \vec{v'} = Q' \vec{v}$

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?