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Space of tensor shape

So, to review some linear algebra. A symmetric matrix can always be diagonalized, that is, decomposed into a rotation, a diagional matrix, and the rotation inverse. The rotation is made of the eigenvectors, and the diagonal matrix is made of the eigenvalues. And this is the natural decomposition of the tensor quantity into orientation and shape information.

And here is how that information is shown in the standard ellipsoid glyph. The orientation of the ellipsoid axes conveys the eigenvectors, and the scaling along the axes gives the eigenvalues. For the time being, I'm only going to be talking about the shape of these things, and we can identify the shape of the tensor with its three eigenvalues, which live in this space here.

We can restrict this space by ignoring the over-all size, and looking at this domain in which the sum of the eigenvalues (the trace of the original tensor) is constant. We can further restrict this domain by looking at the region in which the eigenvalues are sorted, so that we have only one instance of each set of three eigenvalues.