How to calculate easily the eigenmatrix of a 3D tensor. 


I try immersing the tensor in a big matrix, in my case, the tensor is of nxnxn and I can build an n^2 x n^2 matrix that contains all the "coefficients" of my original tensor, but my calculations give me more eigenvalues that I have in my original problem (n^2). 


There exist any numerical library for eigenvalues/eigenmatrices of a tensor?



Eigenvalue and Eigenmatrix of a 3D Tensor - How to calculate it?
You ask for the eigenvalues of an $m=3$-order $n$-dimensional tensor $M$. There is no unique definition of the "eigenvalue" $\lambda$ for $m\geq 3$. One frequently used definition is
$$\sum\_{i\_2,i\_3,\ldots i\_m=1}^n M\_{i,i\_2,i\_3,\ldots i\_m}x\_{i\_2}x\_{i\_3}\cdots x\_{i\_m}=\lambda x\_i^{m-1}, \;\;\text{for all}\;\;i=1,2,\ldots n$$
The problem of computing the eigenvalues for $m\geq 3$ is NP-hard, see [All Real Eigenvalues of Symmetric Tensors](https://arxiv.org/abs/1403.3720) (2014). A treatment specifically for $m=3$ is in [A Spectral Theory for Tensors](https://arxiv.org/abs/1008.2923) (2010) and in the [Wikipedia article on higher-order singular value decomposition](https://en.wikipedia.org/wiki/Higher-order_singular_value_decomposition). Algorithms are part of the [MatLab toolbox.](https://www.tensortoolbox.org/hosvd_doc.html)


