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The definition of symmetric second order tensors (e.g. linear operators, bilinear forms) corresponds essentially to their matrix representations being symmetric matrices.

Question: Is there a notion of transpose for higher order multidimensional arrays (e.g. three- or four-dimensional arrays)?

A natural generalization of symmetric tensor for orders $>2$ would be to define a symmetric higher order tensor to be one whose multidimensional array representation is "symmetric".

However, such a definition would require there to exist some notion of transpose/adjoint for higher-dimensional multidimensional arrays (i.e. $>2$).

For example, can we make a correspondence between "symmetric" three-dimensional arrays and homogeneous polynomials of degree three, the same way we have a correspondence between symmetric matrices and quadratic forms? Or between "symmetric" four-dimensional arrays and homogeneous quartic polynomials?

Wikipedia says that there is such a thing as the symmetric algebra over a vector space, which is a sub-algebra of the tensor algebra, and that the elements of this subalgebra (when the characteristic of the underlying field is zero, which is the only case that interests me) are called symmetric tensors, which do in fact correspond naturally to homogeneous polynomials, e.g. symmetric third order tensors and cubic forms, symmetric fourth order tensors and quartic forms:

The space of symmetric tensors of order r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V.

However, in the same way that symmetric second order tensors naturally have symmetric matrix representations, do higher order symmetric tensors also naturally map to some type of multi-dimensional array which is invariant under some type of generalized transpose operation?


marked as duplicate by Chill2Macht, Community Sep 5 '16 at 16:31

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.


Update: The obvious notion for generalization of a transpose operation would be "reflection about the diagonal". However, for higher-dimensional arrays, this concept could no longer work without modification, because reflection about axes is not possible/is not well-defined in Euclidean spaces of dimension greater than 2. We would need to have some standard "hyperplane of indices" to reflect about instead, which reduces to the diagonal of a matrix in the 2-dimensional case.

This might be possible if there is a generalization of Householder matrices to higher-dimensional arrays, since symmetric matrices can apparently be decomposed into the sum of Householder matrices. So perhaps a symmetric higher-dimensional array would be an appropriate decomposition of higher-dimensional Householder arrays -- this is of course pure speculation.

Also the answer to this question might just be "invariance under all non-identity permutations of indices" as indicated to the answer to this question (which I did not find when originally writing this question). Update to the update: This question seems to be a duplicate of this one: How to generalize symmetry for higher-dimensional arrays?. I apologize for the mistake -- the question did not come up in the list of related questions when I was writing this. Anyway the consensus answer for that question is essentially:

A higher-dimensional array is symmetric if and only if it is invariant under all non-identity permutations of its indices.

Note that this does seem to be the natural definition -- (1) it reduces to the notion of symmetric matrix in 2d, and (2) it corresponds almost exactly to the definition of symmetric tensor (see Wikipedia for this). A major downside to this is that it is somewhat difficult to imagine visually (even in the 3d case), in contrast to the 2d case. In particular, I was hoping for a definition which could be reduced to the symmetry of all sub-matrices of the array, since it is somewhat easier to think about multidimensional arrays in terms of two-dimensional "matrix slices".

As for whether such objects actually exist in higher dimensions, the answer is clearly yes: just take the array with all $1$'s as entries -- this clearly satisfies the definition. As for whether any non-degenerate multi-dimensional arrays satisfy the definition -- that I do not really know.

Anyway I am voting to close this question since it is clearly a duplicate.


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