Are the eigenvectors and eigenvalues of a tensor covariant?

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A lot of online searching hasn't turned up a direct answer to this question. Evidently the eigenvalues and eigenvectors of a tensor are unique (up to a scale factor), so I imagine they would transform the same way the tensor does. However, I haven't found a clear statement to that effect. I'm hoping someone can provide a proof, or provide a link to a source of a proof, one way or the other.

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1 Answer

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Well, I guess it is not clear, what do you mean by eigenvalues of a tensor. One can assign an eigenvalue to a matrix or to an operator.

If your tensor is once covariant and once contravariant, then it acts on vectors as a matrix or an operator and transforms simply as if you changed the basis for the operator. So, the eigenvalues are preserved (i.e. transform as scalars) and the eigenvectors transform as vectors.

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