Timeline for An inequality on some pairs of orthogonal vectors
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| Oct 26, 2020 at 3:46 | answer | added | David E Speyer | timeline score: 9 | |
| Oct 21, 2020 at 16:35 | comment | added | David E Speyer | Here is an approach I have been messing with: $\sum |a_{ij}| = \max_S \ Tr(A S)$ where $S$ ranges over $\pm 1$ matrices (or matrices which are $\pm 1$ off the diagonal). If $A$ is skew symmetric, then we can assume that $S$ is skew symmetric. $|A|_{(1)} = \max_Q Tr(AQ)$ where $Q$ ranges over orthogonal matrices. Can we show that every $\pm 1$ matrix is in the convex hull of $\tfrac{2n}{3} \ O(n)$? | |
| Oct 21, 2020 at 16:25 | comment | added | David E Speyer | Has anyone tried thinking about this for skew-symmetric $A$? It might be easier, since then the condition of having $0$ diagonal is automatic. | |
| Oct 2, 2020 at 19:22 | answer | added | fedja | timeline score: 9 | |
| Oct 2, 2020 at 17:12 | answer | added | Ivan Meir | timeline score: 2 | |
| Oct 1, 2020 at 18:49 | comment | added | fedja | @GiorgioMetafune Yes, and $\pi$ on the plane is sharp, so the planar example is also possible in any dimension, but the proof doesn't generalize. | |
| Oct 1, 2020 at 18:27 | comment | added | Giorgio Metafune | @fedja I understood from your previous comment that the inequalitiy hods with $\pi$ in the plane and you wonder whether this is true in any dimension. Did I understand correctly? | |
| Oct 1, 2020 at 18:02 | comment | added | fedja | @GiorgioMetafune As I said, just take huge $n$ and $a_i$ equidistributed on the unit circle (and $b_i$ to be orthogonal to $a_i$ of length $1$ too). By now I'm up to $1+\sqrt 3$ but it is still short even of the requested $3$, forget the optimal $\pi$. | |
| Oct 1, 2020 at 12:00 | history | edited | Mahdi - Free Palestine | CC BY-SA 4.0 |
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| Sep 30, 2020 at 18:55 | comment | added | Giorgio Metafune | Is there any numerical evidence that $\pi$ is optimal? | |
| Sep 29, 2020 at 20:32 | answer | added | Iosif Pinelis | timeline score: 10 | |
| Sep 29, 2020 at 19:33 | history | edited | Mahdi - Free Palestine | CC BY-SA 4.0 |
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| Sep 29, 2020 at 19:25 | history | edited | Mahdi - Free Palestine | CC BY-SA 4.0 |
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| Sep 29, 2020 at 18:49 | comment | added | Mahdi - Free Palestine | @IosifPinelis and fedja: 3 is not optimal and 3 just is an underestimate of a numerical experiment. Now, I added a matrice reformulation of the problem. The above problem can be considered as a generalization of this question. For symmetric matrices, we think that 4 is optimal. | |
| Sep 29, 2020 at 18:39 | history | edited | Mahdi - Free Palestine | CC BY-SA 4.0 |
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| Sep 29, 2020 at 14:39 | comment | added | Iosif Pinelis | @fedja : I can only prove it with $4/\sqrt3$ in place of $3$ (don't have the time to type this out now). Your $\sqrt{2+2\sqrt5}$ is better. I think it could be useful if you would present your solution. | |
| Sep 29, 2020 at 13:04 | comment | added | fedja | @IosifPinelis Good question; I wonder it too. On the plane (i.e., when $k=2$) $3=\pi$. It looks to me like the uniform distribution on the 2-dimensional circle is the worst case scenario in any dimension, but I cannot prove it. $\sqrt{2+2\sqrt 5}$ instead of $3$ is easy. That's all I currently know. What can you prove? | |
| Sep 29, 2020 at 11:50 | comment | added | Iosif Pinelis | @Mahdi : Can you let us know how this conjecture arose? Also, do you know nontrivial cases when the equality is attained? | |
| Sep 28, 2020 at 7:17 | comment | added | Mahdi - Free Palestine | @JochenWengenroth: Yes, but in that case we don't use conditions $a_i^T b_i = 0$. | |
| Sep 28, 2020 at 7:12 | comment | added | Jochen Wengenroth | Using $2|a^Tb|\le \|a\|^2+\|b\|^2$ you get the inequality with a factor $2/(n-1)$ instead of $3/n$. | |
| Sep 28, 2020 at 6:53 | comment | added | Mahdi - Free Palestine | @abx: thanks, it was edited. | |
| Sep 28, 2020 at 6:51 | history | edited | Mahdi - Free Palestine | CC BY-SA 4.0 |
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| Sep 28, 2020 at 6:48 | comment | added | abx | You probably want $a_i^Tb_j$ in the right hand side sum. | |
| Sep 28, 2020 at 6:30 | history | asked | Mahdi - Free Palestine | CC BY-SA 4.0 |