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One number $a$ can be seen as a one-dimensional matrix. Can we generalize matrices in a high-dimension sense? Think of a “cubic matrix”, which looks like a crystal with a number attached to its vertices. But how to define an appropriate multiplication leaves me a big challenge. I suppose that multiplication between two “cubic matrices”(3d matrix) is done by choosing two surfaces and conducting the normal 2-dimension matrix multiplication.

I wonder the attemp to define a hypermatrix is meaningful or not if the manifestations of hypermatrices might be reduced to 2-dimensional matrix. I am also curious about how to define an appropriate multiplication between hypermatrices if the introduction of hypermatrix is meaningful.

Since now, I have only learned tensor product between two matrices which might share some similarities to my questions. Another motivation raising my question is to generalize the expressions of linear equations $AX=B$ to a high-dimension way.

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  • $\begingroup$ Matrices serve as a way to represent linear transformations of vector spaces. What would your hypermatrices mean, or represent? $\endgroup$ – Allawonder May 20 at 9:23

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