[help-3dldf] Fwd: Re: Testing for ellipsoidality

[Laurence Finston]

------ Forwarded message -------


From: Hans Aberg <address@hidden>
To: Laurence Finston <address@hidden>
Date: Mon, 22 Nov 2004 18:59:11 +0100

If you already know that you have an ellipsoid, and want to check that it
remains so under a linear transformation, then it suffices to check that the
dterminant of the the linear transformation is /= 0: The ellipsoid an its
interior is compact, and so the image is also compact. And mapped under a
linear transformation, it remains of degree at most 2. Now the absolute
value of the determinant of the differential of a (differentiable) function
is the local volume change. So if the determinant is 0, the ellipsoid is
mapped onto something of volume zero, else something of volume /= 0.

If you work with linear transformations, then it is often useful to reduce
using Jordan's normal form of matrices: For any (single) given matrix A, one
can find a basis in which it appears with Jordan blocks on the diagonal.
Each Jordan block is a small k_i x k_i matrix with an eigenvalue lambda_i
one the diagonal, 1's immediately above the diagonal, and 0's elsewhere.
Each lambda_i cabn appaer in different blocks. The different boocks
correspond to different orthogonal eigenspaces. One gets these eigenvalues
by computing the characteristic polynomial p_A(t) := det(tI - A) of A, and
factoring it.

When working with quadratic forms Q, they transform under a basis change as
B^* Q B not as B^(-1) A B. Therefore, the transformations for which these
two basis chenge transformations agree are those with B^* = B^(-1), and
these are called the orthonormal transformations. These preserve both length
and angle.

  Hans Aberg