Timeline for Examples of naturally occurring Quadratic forms or quadrics.
Current License: CC BY-SA 2.5
20 events
| when toggle format | what | by | license | comment | |
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| Sep 4, 2015 at 13:26 | answer | added | Vesselin Dimitrov | timeline score: 3 | |
| Sep 4, 2015 at 11:56 | answer | added | Joe Silverman | timeline score: 6 | |
| Sep 4, 2015 at 10:10 | answer | added | user62562 | timeline score: 2 | |
| Sep 4, 2015 at 9:39 | answer | added | Oblomov | timeline score: 5 | |
| Sep 10, 2013 at 10:26 | history | edited | user9072 |
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| Jun 18, 2011 at 20:30 | comment | added | Vectornaut | Shouldn't the Roman numeral for zero be ( )? ;) | |
| Mar 19, 2011 at 6:49 | answer | added | Jack Rousseau | timeline score: 0 | |
| Mar 18, 2011 at 23:19 | vote | accept | Olivier Bégassat | ||
| Mar 20, 2011 at 0:04 | |||||
| Mar 18, 2011 at 23:10 | comment | added | Olivier Bégassat | You are right about the dimension restriction, it is arbitrary in the case of linear maps. And I am not looking for small dimensional examples in general, I just gave a list of some particular quadratic forms encountered in the litterature. I don't know how I forgot about the Killing form. | |
| Mar 18, 2011 at 22:16 | answer | added | Georges Elencwajg | timeline score: 7 | |
| Mar 18, 2011 at 22:14 | comment | added | KConrad | I would consider as examples that give incentive to study general quadratic forms those constructions which, in general, produce quadratic forms in potentially any number of variables, not just a small number because you fixed some parameter to force a quadratic form. For instance, the trace form on an associative algebra or the Killing form on a Lie algebra. These are symmetric bilinear forms in possibly many variables (depends on the dimension of the algebra) and correspond to some quadratic forms in many variables. | |
| Mar 18, 2011 at 22:11 | comment | added | KConrad | What do you mean by "arise naturally"? In your examples you are often introducing constraints on a dimension to force a quadratic form in what otherwise would really be better considered as a homogeneous polynomial of some degree. For instance, the determinant on endomorphisms of an n-dim. space is a homogeneous polynomial of degree n in n variables. To say the determinant becomes a quadratic form when you set n = 2 seems, to me, to be missing the big picture of what happens in general, since the determinant is usually not a quadratic form. (Continued...) | |
| Mar 18, 2011 at 21:10 | comment | added | Will Jagy | People are quite fond of : $$ $$ Symmetric bilinear forms [by] J. Milnor [and] D. Husemoller by John Willard Milnor, 1973,Springer-Verlag edition, in English. | |
| Mar 18, 2011 at 21:05 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Mar 18, 2011 at 20:41 | comment | added | Olivier Bégassat | All of the examples I listed carry geometric information about the underlying space. The notion of polarity with respect to a quadric can be used to construct new points from old ones naturally. For instance polarity with respect to the radical in the space of quadrics has geometrical meaning (unfortunately I don't have my books nearby to give you the full picture). Also, these examples of natural quadratic forms give incentive to study general quadratic forms, and make for beautiful applications of the general theory. | |
| Mar 18, 2011 at 20:34 | history | edited | Olivier Bégassat | CC BY-SA 2.5 |
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| Mar 18, 2011 at 20:34 | comment | added | Qiaochu Yuan | I think the question is a little broad. Can you be more specific about what you are looking to gain by having such a list? | |
| Mar 18, 2011 at 20:32 | comment | added | Olivier Bégassat | This is indeed naturally associated. But I am more interested in quadrics associated to a space, like the examples I listed above. | |
| Mar 18, 2011 at 20:27 | comment | added | Qiaochu Yuan | Second-order approximation to a potential function at a critical point? | |
| Mar 18, 2011 at 20:10 | history | asked | Olivier Bégassat | CC BY-SA 2.5 |