Timeline for Trace of a finite hypercubic tensor
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 22, 2019 at 6:48 | vote | accept | Luca Cappelletti | ||
| Oct 22, 2019 at 4:15 | comment | added | Michael Engelhardt | Since you request it, I've posted an answer, although it should be said that this is rather off-topic for mathoverflow, which is for research-level questions in mathematics. Yes, the book you link to is the one I meant. | |
| Oct 22, 2019 at 4:14 | answer | added | Michael Engelhardt | timeline score: 1 | |
| Oct 21, 2019 at 20:28 | comment | added | Luca Cappelletti | Thanks, may I ask for you to formulate an answer based on these comments? I believe it could be helpful to a number of peoples with doubts similar to mines (people who I consulted with before asking it here). Additionally, may I ask if this is the book you mentioned? Thanks! | |
| Oct 21, 2019 at 18:03 | comment | added | Michael Engelhardt | 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. | |
| Oct 21, 2019 at 13:58 | comment | added | Luca Cappelletti | 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? | |
| Oct 21, 2019 at 13:33 | comment | added | Michael Engelhardt | 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. | |
| Oct 21, 2019 at 6:46 | comment | added | Luca Cappelletti | 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! | |
| Oct 21, 2019 at 2:56 | comment | added | Michael Engelhardt | 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. | |
| Oct 20, 2019 at 14:56 | history | asked | Luca Cappelletti | CC BY-SA 4.0 |