In matrix theory(2-dimensional arrays), we can define addition, multiplication, rank and determinants etc.


I'm working on generalizing these properties to multidimensional arrays as many as possible. Are there any results in this direction? I'd really appreciate it if you could provide some references.



Are there any results in generalizing matrix theory to multidimensional arrays?
**Yes, many of them**.


The keyword you should look for is *tensors* (note that it is used in a slightly different meaning in the physics literature, though).


I suggest to start from [Kolda and Bader's 2009 SIREV review paper](https://www.sandia.gov/%7Etgkolda/pubs/pubfiles/TensorReview.pdf), for instance, or from [Van Loan's lecture notes](https://doi.org/10.1007/978-3-319-49887-4_1) here.


