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Is the trace of a finite hypercubic tensor defined?

Clearly, for the bidimensional case $n \times n$ the trace is defined as the sum of the elements on the main diagonal:

$$\operatorname {tr} (\mathbf {A}_{2D} )=\sum _{i=1}^{n}a_{ii}=a_{11}+a_{22}+\dots +a_{nn}$$

Is there any sound attempt at generalizing the trace to hypercubic tensors of shape $n\times n \times \ldots \times n$?

I can imagine for instance something like:

$$\operatorname {tr} (\mathbf {A}_{mD} )=\sum _{i=1}^{n}a_{\mathbf{i}}=a_{\mathbf{1}}+a_{\mathbf{2}}+\dots +a_{\mathbf{n}}$$

where $\mathbf{i}$ is $m$ times $i$.

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    $\begingroup$ I fear your question disregards the characterizing property of a tensor, namely, its transformation behavior. You can always contract two indices of a tensor and obtain again a tensor; the fact that for a rank-2 tensor, this happens to look like a trace is an accident of representing the tensor as a matrix and doesn't mean that the trace is an operation which it necessarily makes sense to generalize for tensors. Indeed, the generalization you suggest doesn't seem to have any sort of defined transformation behavior. It's not a scalar, like the rank-2 version is. $\endgroup$
    – Michael Engelhardt
    Commented Oct 21, 2019 at 2:56
  • $\begingroup$ Sure, it is meant to work only on hypercubic tensors, I understand. Why shouldn't it be a scalar? The vector in the second formula are only the indices, so it would be the sum of the hyper-diagonal. I'm pretty ignorant of this kind of algebra (computer science PhD, not nearly enough math), so any help would be very welcome indeed. Thanks! $\endgroup$
    – Luca Cappelletti
    Commented Oct 21, 2019 at 6:46
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    $\begingroup$ Work out what happens to your expression if you rotate the basis, say, in the 1-2 plane. It's not a scalar. The mistake you're making is that you're conflating tensors and matrices. These are conceptually very different. Just because tensors can be (don't have to be!) represented as matrices doesn't mean that every concept that one has for matrices can be meaningfully applied to tensors. Just because birds are a manifestation of an animal doesn't mean that one can sensibly talk about "an animal's wings" in general. $\endgroup$
    – Michael Engelhardt
    Commented Oct 21, 2019 at 13:33
  • $\begingroup$ Okay, thank you! Would you say that the generalization I've proposed would apply in the case of multi-dimensional matrices (not tensors)? Also, would you suggest any good explanation of the differences between matrices and tensors? $\endgroup$
    – Luca Cappelletti
    Commented Oct 21, 2019 at 13:58
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    $\begingroup$ The concepts of matrices and tensors should be explained in any decent applied mathematical methods book, say, Riley, Hobson and Bence. It's certainly possible to define the object you suggest for multi-dimensional arrays of numbers (matrices, if you wish) - it's not immediately obvious what an application would be. $\endgroup$
    – Michael Engelhardt
    Commented Oct 21, 2019 at 18:03

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Although this is rather off-topic for mathoverflow, I'll provide an answer, given that no objections have been raised to the OP and since it's being explicitly requested:

First, one should carefully distinguish between tensors and matrices. These are distinct concepts, each with their own properties and with their own sets of operations that can be meaningfully carried out on them. Although contact points between these concepts exist, e.g., when one represents tensors in terms of matrices, one should not attribute properties of the one to the other.

The characterizing property of a tensor is its transformation behavior. This, in particular, allows for the "contraction" operation - setting two indices equal and summing over them. This operation produces again a tensor. By contrast, setting more than two indices equal and summing over them, as suggested in the OP, or even just summing over only one index does not produce again a tensor and therefore is not a useful operation on tensors.

Matrices (including higher dimensional versions as alluded to in the OP) are a priori a more malleable concept. In principle, they're just a way of arranging information, and it's usually necessary to explain what that information is in addition to writing the matrix itself, just as the components of a vector have no meaning without an explanation of what basis they are associated with. The malleability of the concept implies that one can define operations on matrices fairly freely. In principle, one can define the operation suggested in the OP as a generalization of the "trace" operation on $n\times n$ matrices - this by itself, however, does not yet impart meaning to the operation or suggest a possible application.

The tensor and matrix concepts make contact if one chooses to represent tensors as matrices. Different choices of bases lead to different matrix representations of a given tensor, underscoring again the distinction between the concepts. It so happens that the contraction of a rank-2 tensor corresponds to taking the trace of any of its $n\times n$ matrix representations (in a slight abuse of language, one might sometimes speak of the trace of the tensor). In particular, the result is independent of representation - it is a scalar. This property does not persist for the generalizations suggested by the OP, which do not again lead to a tensor. Thus, an application of these generalizations in the tensor context seems unlikely. It therefore remains unclear what a possible application might be - similar to the notion of just adding up the components of a vector, which also is an operation that, at the very least, requires further explication to be meaningful.

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