Timeline for Prove/disprove a linear algebra inequality
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| S Jun 12, 2023 at 5:03 | history | bounty ended | Chen Zeno | ||
| S Jun 12, 2023 at 5:03 | history | notice removed | Chen Zeno | ||
| Jun 12, 2023 at 5:03 | vote | accept | Chen Zeno | ||
| Jun 11, 2023 at 23:10 | answer | added | jmd | timeline score: 5 | |
| Jun 11, 2023 at 10:22 | answer | added | user42355 | timeline score: 8 | |
| Jun 11, 2023 at 6:53 | comment | added | Chen Zeno | Thank you for your comment. Yes, we want a detailed answer if possible. Yes, we have a proof for the special case where $a=\pm w$. | |
| Jun 10, 2023 at 5:47 | comment | added | user42355 | I have a proof, but I vary $a$ and $w$, not the $u_j$'s. Should I post it in a detailed answer? Or do you only want hints? (You have asked for "any suggestions", so I am not sure.) E.g., can you prove the inequality in the special case $a = \pm w$? (This is a step in my proof.) | |
| S Jun 7, 2023 at 8:01 | history | bounty started | Chen Zeno | ||
| S Jun 7, 2023 at 8:01 | history | notice added | Chen Zeno | Draw attention | |
| Jun 7, 2023 at 8:00 | history | edited | Chen Zeno | CC BY-SA 4.0 |
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| Jun 5, 2023 at 5:41 | comment | added | Daniel Soudry | Thanks @Echo. We tried your approach. It worked for the first iteration (when the local maximum is an interior point) from the stationarity condition of the gradient. However, then we got stuck at second iteration (when one of the constraints is active). So now we have a stationarity condition on the projected gradient. We couldn't solve this. Any help would be appreciated (I'm working with Chen Zeno). | |
| May 14, 2023 at 10:45 | comment | added | user473423 | Consider the expression as a function on $u$. Use differentials to see that it has no local maximum. Then the maximum must be taken where at least two are perpendicular. Iterate until they are pairwise perpendicular in which case the estimate (with $\le$) is trivial. | |
| May 13, 2023 at 18:58 | history | edited | Chen Zeno | CC BY-SA 4.0 |
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| May 12, 2023 at 12:36 | history | edited | LSpice | CC BY-SA 4.0 |
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| May 12, 2023 at 8:30 | comment | added | Chen Zeno | Thank you for your comment. Your example is correct, but we want to prove this statement for $\{u_i\}_{i=1}^{n}$ such that w exists | |
| May 12, 2023 at 8:10 | comment | added | Beni Bogosel | Are you sure a vector $w$ with the requested property exists? For example if you have two vectors $u_1=-u_2$ then no such $w$ exists. Basically you want the angles between all the $u_i$ to be bigger than $\pi/2$ and a vector $w$ which makes an acute angle with all of them. (Edit: maybe in higher dimension this is possible. Take an acute cone around $w$ and distribute evenly some vectors $u_i$ such that the pairwise angles are bigger than $\pi/2$) | |
| May 12, 2023 at 7:46 | history | edited | Chen Zeno | CC BY-SA 4.0 |
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| May 12, 2023 at 7:37 | history | asked | Chen Zeno | CC BY-SA 4.0 |