I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, *hypermatrices*.
Of course multidimensional arrays are useful:
every programming language supports them, and I often employ them myself.
But these uses treat the arrays primarily as convenient data structures
rather than as mathematical objects.
When I think of the generalization of polygon to $d$-dimensional polytope,
or of two-dimensional surface to $n$-dimensional manifold,
I see an *increase* in mathematical importance and utility;
whereas with matrices, the opposite.

One answer to my question that I am prepared to acknowledge is that
my perception is clouded by ignorance:
hypermatrices *are* just as important, useful, and prevalent in mathematics
as 2D matrices.
Perhaps tensors, especially when viewed as multilinear maps, fulfill
this role. Certainly they play a crucial role in physics,
fluid mechanics, Riemannian geometry, and other areas.
Perhaps there is a rich spectral theory
of hypermatrices, a rich decomposition (LU, QR, Cholesky, etc.) theory
of hypermatrices, a rich theory of random hypermatrices—all
analogous to corresponding theories of 2D matrices, all of which I am unaware.

I do know that Cayley explored hyperdeterminants in the 19th century, and that
Gelfand, Kapranov, and Zelevinsky wrote a book
entitled *Discriminants, Resultants and Multidimensional Determinants*
(Birkhäuser, Boston, 1994) about which I know little.

If, despite my ignorance, indeed hypermatrices have found only relatively rare utility in mathematics, I would be interested to know if there is some high-level reason for this, some reason that 2D matrices are inherently more useful than hypermatrices?

I am aware of how amorphous is this question, and apologize if it is considered inappropriate.

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