I spent time today talking to Kate Storey-Fisher about features to use in her cosmological regression projects. The point is that we are only considering features that have well-defined, coordinate-free meanings, because we are trying to do regressions that are invariant to coordinate transformations. These features include scalars, vectors, and tensors, which we contract into scalars. But what can you do with a tensor? At order 2, a tensor has a trace (its self-contraction); it can be contracted with two vectors; it has eigenvalues and eigenvectors. The eigenvalues are classical scalars; good! But are the eigenvectors classical vectors? No, they aren't, because they don't have signs. What can you do with them? I have some theories...
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