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I am working with hypergraphs. The various matrices associated with hypergraphs are hypermatrix or tensors. I am interested in spectral aspects. In particular, I want to find all the eigenvalues explicitly for a class of hypergraphs. To start with, we can consider all the $0-1$ hypermatrices of order $2 \times 2 \times 2$. Then the characteristic polynomial is defined in terms of resultants of a certain system of homogeneous equations.

I want to know, some simple or at least a concrete way of calculating these eigenvalues. I have understood the method given in this paper. But this method was applicable only to special hypermatrices. Kindly share some references. Thank you.

Edit: I am interested in general uniform hypergraph then its associated matrices can be of any order and dimension. I thought, to start with, I will look at the simplest $2 \times 2 \times 2$ case.

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    $\begingroup$ See mathoverflow.net/questions/319644/…. See also stat.uchicago.edu/~lekheng/work/mcsc2.pdf, discusses eigenvalues of hypergraphs starting on slide 22. See also mathscinet.ams.org/mathscinet-getitem?mr=1325271 (but I don't have easy access to the actual article, so I'm not sure how relevant it really is). $\endgroup$
    – Zach Teitler
    Commented May 12, 2020 at 5:54
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    $\begingroup$ Despite the above comment and already accepted answer, I think there might still be something to say here, and if some expert would care to step in, it would be most welcome. I'm a little fuzzy on this but it seems to me that a symmetric $2 \times 2 \times \dotsm 2$ tensor ($d$ factors) corresponds to a binary $d$-form, and the eigenvectors correspond to that form's critical points (on the projective line), ie., roots of the derivative (of a dehomogenization). The eigenvalues would presumably be the critical values, but I might be off by a scalar factor (???)... $\endgroup$
    – Zach Teitler
    Commented May 13, 2020 at 3:35
  • $\begingroup$ ... A scalar factor is not an issue at multiple roots of the original form (eigenvalue $0$) but for the others I would have to think about it. Well, I don't know how interesting hypergraphs on $2$ vertices are, but as far as eigenvalues of $2 \times 2 \times \dotsm \times 2$ tensors, I think that it should fall out easily from simple things with binary forms (even if I have some of the details wrong). $\endgroup$
    – Zach Teitler
    Commented May 13, 2020 at 3:38
  • $\begingroup$ @ZachTeitler It is very interesting thank you. Can you share some references? Also, how to see that the eigenvectors correspond to that form's critical points (on the projective line) and eigenvalues are critical values? Can you share some examples as an answer also? I am interested in general uniform hypergraph whose associated matrices can be of any order and dimension. I thought, to start with, I will look at the simplest $2 \times 2 \times 2$ case. $\endgroup$
    – GA316
    Commented May 13, 2020 at 11:33

1 Answer 1

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As explained in a previous MO question, there is no unique generalization of the eigenvalue of an $n\times n$ matrix to an $n\times n\times n$ tensor. One approach is to construct a higher-order singular value decomposition. This has been worked out for the specific case of $2\times 2\times 2$ tensors by Ana Rovi in a M.Sc. thesis. Even for this simple case, there is no simple closed-form expression. A MatLab toolbox may be useful to implement the procedure.

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    $\begingroup$ Thank you for your answer and wonderful references. $\endgroup$
    – GA316
    Commented May 12, 2020 at 19:33

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