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Just as eigenvectors are invariant vectors up to a multiplicative constant $Au=\lambda u$, I wondered if eigentensor theory is equally developed for any arbitray tensor or hypermatrix array, i.e., $AB=\lambda B$, where A and B are suitable hyperobjects, and, maybe, even $\lambda$ could be an eigentensor as well.

Related: Why should we encounter eigentensors naturally? Is the eigentensor/hypermatrix theory hard? Any nice reference for current status of the subject including something else beyond hyperdeterminants (Kapranov et alii)?

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  • $\begingroup$ You might be interested in the theory of eigenoperators. $\endgroup$
    – Cameron Williams
    Jan 28, 2018 at 19:13
  • $\begingroup$ That thing...Does really exist? LoL Academia is obsolete, really! ;) Any reference? $\endgroup$
    – riemannium
    Jan 28, 2018 at 19:29
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    $\begingroup$ It could also be called an eigenvector, just in a "vector space" of tensors. For example, on the space of $n \times n$ real matrices ${\text M}_n(\mathbf R)$ the transpose operation $B \mapsto B^\top$ is a linear operator of order 2 and the symmetric matrices are all eigenvectors of this operator. The function $e^{\lambda x}$ is an eigenvector of the differentiation operator $d/dx$, with eigenvalue $\lambda$, although in the setting of analysis these are called eigenfunctions instead of eigenvectors, but it's the same thing. $\endgroup$
    – KCd
    Jan 28, 2018 at 20:26
  • $\begingroup$ @riemannium I've only seen it scattered across papers, but not in books. I'd suggest searching around and finding papers that seem relevant. $\endgroup$
    – Cameron Williams
    Jan 28, 2018 at 21:05

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The answer to both questions is yes. The theory is very new, but here are two textbooks you could explore:

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