Timeline for Geometric interpretation of characteristic polynomial
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Mar 20, 2019 at 21:44 | answer | added | Sean Lawton | timeline score: 11 | |
| S Aug 13, 2014 at 5:56 | history | suggested | Ali Taghavi |
I added a tag, since these coefficients are called invariant polynomials and the corresponding $G$ is $GL(n,\mathbb{R}$
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| Aug 13, 2014 at 5:39 | review | Suggested edits | |||
| S Aug 13, 2014 at 5:56 | |||||
| Jul 27, 2010 at 7:35 | vote | accept | Per Vognsen | ||
| Jul 27, 2010 at 7:31 | comment | added | Pete L. Clark | @PV: What you say both agrees with what I said (or meant to say, at least) and goes on to give a geometric interpretation of the sort I vaguely had in mind. Cheers. | |
| Jul 27, 2010 at 7:21 | answer | added | Ryan Budney | timeline score: 77 | |
| Jul 27, 2010 at 7:20 | comment | added | Per Vognsen | Pete: The exterior powers of the operator aren't mysterious to me. But if you apply the interpretation of the trace directly to this question, it will give you an answer in terms of the various $\wedge^p(V)$ vector spaces rather than $V$ itself. Anyway, I think I got it: If we take $R^3$ with a diagonal matrix $diag(a_1, a_2, a_3)$ as an example, the bivector trace is $a_1 a_2 + a_2 a_3 + a_1 a_3$. This is the second-order volume differential contributed by the edges of the unit cube, much as the vector trace $a_1 + a_2 + a_3$ is the first-order volume differential contributed by the faces. | |
| Jul 27, 2010 at 7:02 | comment | added | Pete L. Clark | You seem very confident that the trace has a geometric interpretation; in fact, this was the subject of a previous MO question mathoverflow.net/questions/13526/…. But if you are truly happy with the geometricity of the trace, it seems that your question comes down to asking for a geometric interpretation of "intermediate" exterior powers of a linear operator. I'm sure that some people here would be happy to speak to that... | |
| Jul 27, 2010 at 6:49 | comment | added | Victor Protsak | This formula is certainly mentioned in more advanced books that take coordinate-free point of view. | |
| Jul 27, 2010 at 6:36 | history | asked | Per Vognsen | CC BY-SA 2.5 |