# Is there a generalization of eigenvalues and eigenvectors to tensors?

Two perhaps ill-posed or just silly questions:

1. Let $n>0$, $T$ be an $(n+2)$-tensor, and $\otimes$ denote the Kronecker product of tensors. Is there a tensor generalization for the fundamental eigenvector-eigenvalue relationship $M\cdot v_k - \lambda_k v_k = 0$ of the form: $$T \otimes T'_k = T''_k \otimes T'_k,$$ where $T'_k$ is an $(n+1)$-tensor, $T''_k$ is an $n$-tensor, and $k\in\mathbb Z^{n+1}$?
2. If [1] is not silly, does this structure form a hierarchy for decomposing $T$ into sums/products of such $(n+1)$- and $n$-tensors $T'_k$ and $T''_k$? My apologies for bad notation. Thank you for your comments.
• you'll really want to make an effort to code your formulas in LaTeX; as it stands, they are only legible with much effort. – Carlo Beenakker Jul 17 '18 at 17:00
• I made some changes to the formatting; if the meaning of your question was changed, sorry about that, and feel free to revert. – Arun Debray Jul 17 '18 at 17:12
• You might want to start with the introductory article by Bernd Sturmfels, ams.org/journals/notices/201606/rnoti-p604.pdf. From there you can follow the references. And you can also search online for terms like "tensor eigenvectors" to find plenty more articles. – Zach Teitler Jul 17 '18 at 17:37
• Sorry, but this question doesn't make sense as currently stated. In (1), $T \otimes T'_k$ is a $(2n+3)$-tensor and $T''_k \otimes T'_k$ is a $(2n+1)$-tensor. They can't be equal. In $Mv = \lambda \otimes v$, the $Mv$ is not a Kronecker product. It's matrix multiplication, or Kronecker product followed by a contraction. – Zach Teitler Jul 17 '18 at 17:42

## 1 Answer

There are several generalizations of the concept of "eigenvalues" to tensors. A good starting point on this topic is this paper by L.-H. Lim