I worked through this problem, so I am posting my work (maybe somebody else would find this informative.)

- - -

I am interested in seeing a relationship for standard ellipses as the eccentricity varies, so there is no rotation and the major axis, a, is 1, and aligned along the x-axis.

So, given the standard form of an ellipse

and knowing that

the equation can be rearranged into the following form:

This means that the diagonal matrix, D, is

Because there is no rotation (the a- and b-axes are aligned with the x- and y-axes), Q, which can be considered a rotation matrix, is

det(Q) is 1, so it does, indeed, serve as a rotation matrix.

The A matrix can now be computed as follows:

which works out to

which is the same as the D matrix.

The equation in its original quadratic form is the same as the one found above:

I had hoped to find some more information but, unfortunately, did not.