But not any multidimensional data is a valid tensor.
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FYI, matrices transform between vectors within a vector space.
You bring up a good point though, if this were meant to be a transformation, then we're talking about modules (Z/2^8Z being the underlying ring) and not vector spaces, which is fine. I was needlessly narrow when I said "vector spaces" earlier.
Square ones do, but m x n ones represent linear maps from an n-dimensional to m-dimensional vector space (over the/a field containing the elements of the matrix).