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Have the 2D matrix operations been generalized to n-dimensional matrices?

Are there any books that define various operations on multidimensional matrix? I'd like to see operations such as multidimensional matrix multiplication, multidimensional matrix transpose, multidimensional matrix inversion.

I got only this reference on the net.

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    $\begingroup$ Don't you mean tensors ? $\endgroup$
    – Piyush Grover
    Commented Jun 18, 2013 at 21:44
  • $\begingroup$ No. I wonder if there could be any differences between tensors and multidimensional matrices. $\endgroup$
    – tem pora
    Commented Jun 18, 2013 at 21:46
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    $\begingroup$ Multidimensional matrices are known as tensors, and working with them is known as multilinear algebra. If you dig around, you can find expositions of this. $\endgroup$
    – Deane Yang
    Commented Jun 18, 2013 at 21:48
  • $\begingroup$ @Yang thank you. I guess, the term multidimensional matrix is not used in mainstream literature. That was the main reason for my confusion. $\endgroup$
    – tem pora
    Commented Jun 18, 2013 at 22:00
  • $\begingroup$ I guess they use n-dimensional matrices in (n+1)-dimensional worlds $\endgroup$
    – Pietro Majer
    Commented Dec 27, 2014 at 14:32

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There are two old books, authored by N.P. Sokolov, and available in Russian only:

"Spatial matrices and their applications", Gos. Izdat. Fiz.-Mat. Lit., Moscow, 1960 [MR: 0130256] and "Introduction to the theory of multidimensional matrices", Naukova Dumka, Kiev, 1972 [MR: 0352115]. At least the first of these books is available on the web.

From the MR review of the first book:

Such arrays were first considered by Cayley [Trans. Cambridge Philos. Soc. 8 (1842/49), 75–88; pp. 85–88] and have since been the subject of numerous investigations. Of the considerable literature that has grown up in this field, we may mention a long series of papers by M. Lecat published between 1910 and 1929 and also, in more recent years, the work of R. Oldenburger.

I guess this topic is out of fashion now.

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  • $\begingroup$ What's in these books that's different from what's known about tensors? $\endgroup$
    – Deane Yang
    Commented Dec 27, 2014 at 17:40
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    $\begingroup$ @Deane Yang: Frankly, I do not know, I haven't looked thoroughly. I think the difference is in the language, e.g. matrices (as rectangular arrays of numbers) vs. linear maps. $\endgroup$
    – Pasha Zusmanovich
    Commented Dec 27, 2014 at 19:53

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