Tensor analog of Matrix Product

Question

Given two $n \times n$ matrices $A$ and $B$, we can form their matrix product in the usual way. Is there a similar product for tensors? E.g., if one is given two $n \times n \times n$ tensors $\mathcal{A}=(a_{ijk})$ and $\mathcal{B}=(b_{ijk})$, is there an $n \times n \times n$ tensor $\mathcal{C}=(c_{ijk})$ that is rightfully called the product of $\mathcal{A}$ and $\mathcal{B}$?

Answer 1

I think it is still an open problem to find an appropriate multiplication between tensors.

But there are definitions of multiplication between a tensor and a matrix or a vector.

Please check this artical

Multilinear operators for higher-order decompositions

by TG Kolda

and

Tensor Decompositions and Applications

by TG Kolda and BW Bader.