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We all have seen matrices that are "bi-dimensional" (i'm using the quotes here because i'm not talking about the number of lines, but about the way you represent a matrix, as a rectangle).

I was wondering if we can define a space where we have similar objects, but instead of being a rectangle it would be a box, cubic or non-cubic. Imagine for a example a space $M$ of "3D" Matrices, and a 3D matrix $A$ would have numbers represented by $a_{ijk}$, where $i$ is equivalent to a column, $j$ to a line, and $k$ to another line, but this time in a 3-dimensional way. We can imagine a simple $2\times2\times2$ "3D-Matrix" we would have eight entries: $a_{111}, a_{112}, a_{121}, a_{122}, a_{211}, a_{212}, a_{222}$ instead of 4 in a normal matrix.

Does this concept exist? I have tried looking in this forum for "3D matrices" but didn't found anything. Also looked on google but couldn't find anything. Thanks.

embedded_dev
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2 Answers2

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In combinatorial circles I have heard these referred to as "hypermatrices". You can define binary addition and an $n$-ary multiplication(where $n$ is the number of dimensions of the hypermatrix.) that is a natural analogue of matrix multiplication.

Unfortunately you lose a lot of theory because, for example, I don't think anyone has found a natural analogue to eigenvalues or many other concepts natural to the study of matrices.

Sean English
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  • Can we multiply hypermatrices and matrices? – embedded_dev Mar 15 '17 at 02:15
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    You can think of an $n\times m$ matrix as an$ n\times m\times1\times\dots\times1$ hypermatrix, so in that sense you can multiply a matrix against hypermatrices. – Sean English Mar 15 '17 at 11:13
  • Interesting. Thanks! – embedded_dev Mar 15 '17 at 15:13
  • These have an application as creating a representation theory for n-ary groups. Much like matrices over rings give you a representation theory for normal groups. I'm still learning about what the status of 3-ary and higher representation theories looks like – Sidharth Ghoshal Nov 15 '20 at 17:57
  • You can probably find eigenvalues too its worth noting that if you have a 3-index object $a_{ijk}$ then it can act on a a 2-index object $A$ to produce a 1-index object $A'$ and it can act on a 1-index object $B$ to product a 2-index object $B'$. So a natural eigen value problem is to find $A,B$ such that there are constants $c_1, c_2$ such that $a_{ijk}A = c_1 B$ and $a_{ijk}B = c_2 A$. – Sidharth Ghoshal Jun 07 '22 at 00:21
  • When you have object with an even number of indices (such as matrices) this looks more like a typical eigen-value problem, but when the number of indices is odd it seperates into an eigen system. (There's in fact a spectrum of eigenvalue type problems you can recover from here) – Sidharth Ghoshal Jun 07 '22 at 00:23
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Yes, these exist. "3D array" would be a common, essentially unambiguous way to refer to this in computer science. As NeedForHelp commented, sometimes "tensor" is used for an "$n$D array", but tensor has a few different closely-related meanings, so I would be careful with that one.

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Mark S.
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