Maths - Hypermatrix

Since vectors can be thought of as a 1 x n array of elements, and a matrix is an n x m array of elements, then why not continue this sequence to mathematical objects with n x m x p elements and n x m x p ...

hypercube

What would be the multiplication rules for such an algebra?

I can't think of an application for this (unless it is the same as a tensor?) in which case it is used extensively in electromagnetic fields, relativity, quantum mechanics and particle physics.

Could it be, for instance, by analogy with a linear equation, where a matrix transforms on vector into another:

Vout 0
Vout 1
Vout 2
Vout 3
=
m00 m01 m02 m03
m10 m11 m12 m13
m20 m21 m22 m23
m30 m31 m32 m33
Vin 0
Vin 1
Vin 2
Vin 3

that a hypermatrix might transform one matrix into another (multi-linear algebra?):

mo00 mo01 mo02
mo10 mo11 mo12
mo20 mo21 mo22
= hypercube
mi00 mi01 mi02
mi10 mi11 mi12
mi20 mi21 mi22

Of course we can transform a 3x3 matrix into another 3x3 matrix by multiplying it by a third 3x3 matrix. Would it be more general to multiply it by a hypermatrix?

Or is it possible to have a matrix equivalent of multivectors (Clifford algebra) with two types of multiplication. An inner product which transforms:

real number <-- vector <--- matrix <-- hypermatrix

and an outer product which transforms:

real number --> vector ---> matrix --> hypermatrix

What is the relationship between a hypermatrix and a tensor?

Is a hypermatrix just a representation of a tensor?

I think that perhaps a tensor must be square, cuboid, ... In other words it must be the same size in each dimention?

So perhaps not all hypermatrices are tensors?

Rank representation element notation
0 scalar a
1 vector ai
2 matrix aij
3 hypermatrix of rank 3 aijk

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cover us uk de jp fr ca Schaum's Outline of Theory and Problems of Tensor Calculus - I'm finding this hard going, it starts off with as review of linear algebra, matrix notation, etc. It redefines a lot of conventions which are hard to relearn, such as superscrips instead of subscripts to identify elements, and a summation convention, then it goes into coordinate transformations. I cant find any definition of what a tensor is. I think this book is aimed at people who already have some knowledge of the subject.

I can't find any mention in this book of the term hyper-matrix.

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