Timeline for Conditions for when an off-centre ellipsoid fits inside the unit ball
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 3, 2013 at 17:42 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Clarified the nature of the arguments and supplied some missing details
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| Sep 3, 2013 at 17:00 | comment | added | Robert Bryant | @Antony: Actually, I have thought about this a little more and see that the statement can be cleaned up a bit and the ambiguity about the positive or negative definite situation can be removed. I'll modify my answer accordingly. | |
| Sep 1, 2013 at 22:25 | comment | added | Antony | In fact, maybe this even simplifies the test for definiteness, since we only require the product $p_1 p_3>0$ (or equivalently the ratio) rather than the stronger individual conditions $p_1>0$ and $p_3>0$? | |
| Sep 1, 2013 at 22:14 | comment | added | Antony | Thanks a lot, that is very helpful. I believe the extension from working out whether $Q(\nu)>0$ to working out whether $Q(\nu)>0$ or $Q(\nu)<0$ is reasonably straightforward. Using your previous notation, let $p_i(\nu)$ denote the determinant of the upper lefthand $i$-by-$i$ minor. Then $Q(\nu)>0$ if and only if $p_i(\nu)>0$ for $i=1,2,3,4$, whilst $Q(\nu)<0$ if and only if $p_i(\nu)>0$ for $i=2,4$ and $p_i(\nu)<0$ for $i=1,3$. | |
| Sep 1, 2013 at 21:22 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed some typos
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| Sep 1, 2013 at 17:05 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed an error in the answer and added some requested explanations
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| Sep 1, 2013 at 15:53 | comment | added | Robert Bryant | @Anthony: OK, I'll add this as an addendum to the answer. | |
| Sep 1, 2013 at 11:59 | comment | added | Antony | Could you please explain why the original question is equivalent to $Q(\nu)>0$? I can see roughly how the argument would go (we want a vector $\vec{v}$ for which $\vec{v}^\mathrm{T}Q\vec{v}>0$ to achieve $x^2+y^2+z^2-1>0$) but don't quite get it exactly. | |
| Aug 25, 2013 at 17:29 | history | edited | Robert Bryant | CC BY-SA 3.0 |
clarified a sentence
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| Aug 25, 2013 at 17:23 | history | answered | Robert Bryant | CC BY-SA 3.0 |