# From eigenvectors to eigentensors. Is eigentensor (hypermatrix) theory developed and useful?

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)?

• You might be interested in the theory of eigenoperators. Jan 28, 2018 at 19:13
• That thing...Does really exist? LoL Academia is obsolete, really! ;) Any reference? Jan 28, 2018 at 19:29
• 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.
– KCd
Jan 28, 2018 at 20:26
• @riemannium I've only seen it scattered across papers, but not in books. I'd suggest searching around and finding papers that seem relevant. Jan 28, 2018 at 21:05