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