Timeline for Are these two structures isometric?
Current License: CC BY-SA 4.0
39 events
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| Jan 10, 2020 at 20:31 | vote | accept | Ali Taghavi | ||
| S Jan 5, 2020 at 18:21 | history | bounty ended | Ali Taghavi | ||
| S Jan 5, 2020 at 18:21 | history | notice removed | Ali Taghavi | ||
| Jan 4, 2020 at 16:50 | answer | added | Robert Bryant | timeline score: 13 | |
| Jan 3, 2020 at 23:51 | comment | added | Deane Yang | Could you say more about why you find this particular metric interesting and want to study it? Since your questions can in principle be answered by a straightforward calculation, I'm not sure it is a research level question. Can you say more about what prevents you from doing this calculation? | |
| Jan 3, 2020 at 22:53 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jan 3, 2020 at 20:21 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jan 3, 2020 at 19:59 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jan 3, 2020 at 18:45 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| S Jan 3, 2020 at 17:51 | history | bounty started | Ali Taghavi | ||
| S Jan 3, 2020 at 17:51 | history | notice added | Ali Taghavi | Draw attention | |
| Jan 3, 2020 at 12:49 | history | edited | Ali Taghavi |
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| S Aug 12, 2017 at 11:37 | history | bounty ended | CommunityBot | ||
| S Aug 12, 2017 at 11:37 | history | notice removed | CommunityBot | ||
| Aug 4, 2017 at 11:25 | comment | added | Ali Taghavi | @DeaneYang In this metric. the inner product of every two vectors(say the standard vectors) $e_i,e_j$ is equal to $e_i^{tr} (B^{tr})^{-1}B^{-1}e_j$ where $B=A\otimes A$. Now is my question, clear now? | |
| S Aug 4, 2017 at 9:54 | history | bounty started | Ali Taghavi | ||
| S Aug 4, 2017 at 9:54 | history | notice added | Ali Taghavi | Draw attention | |
| Aug 3, 2017 at 19:27 | comment | added | Ali Taghavi | @DeaneYang I think I provided the orthonormal base. For the first metric, it is the columns of $A \otimes A$. Now is the question, clear? | |
| Jul 31, 2017 at 22:48 | comment | added | Deane Yang | You still have not provided an orthonormal basis of tangent vectors. Do you intend that $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$ be an orthonormal basis of $T_AGL(n)$? | |
| Jul 31, 2017 at 22:16 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Jul 31, 2017 at 20:58 | comment | added | Ali Taghavi | @DeaneYang Could I clarify the question after the last edit?Thanks again for your attention to my question? | |
| Jul 31, 2017 at 20:53 | comment | added | Deane Yang | I'm sorry but I won't be able to help any more with this. | |
| Jul 31, 2017 at 19:54 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Jul 31, 2017 at 19:26 | comment | added | Ali Taghavi | @WillieWong Thank you for your attention on my question. My apologies for my delay in response. I add an edit. The delay was because i did not access to computer yester day but i had accecc to my phon and typing was difficult. | |
| Jul 31, 2017 at 19:22 | comment | added | Ali Taghavi | @DeaneYang My apologies for my delay in response.I explained the identificatiom. | |
| Jul 31, 2017 at 19:20 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Jul 31, 2017 at 4:44 | review | Close votes | |||
| Jul 31, 2017 at 9:25 | |||||
| Jul 31, 2017 at 4:25 | comment | added | Deane Yang | Ali Taghavi, I still don't understand your replies. It would be very helpful if, given a matrix $A = [ a_{11} a_{12} ; a_{21} a_{22}]$, you could provide explicit formulas for the 4 columns of $A\otimes A$. | |
| Jul 31, 2017 at 3:00 | comment | added | Willie Wong | @AliTaghavi: please give an explicit example. Quite possibly the identification you have in mind for identifying $2\times 2$ matrix $(a,b; c,d)$ with a vector in $\mathbb{R}^4$ sends the matrix to $(a,b,c,d)$. But it could equally well send to $(a,d,c,b)$ or any other combination. The identification can even depend on your position on $GL(n,\mathbb{R})$. That piece of information maybe in your head, but it is not given to us. | |
| Jul 30, 2017 at 23:05 | comment | added | Ali Taghavi | @DeaneYang we already identify $M_n(\mathbb{R})$ with $\mathbb{R}^{n^2}$. Now please see my previous comment. | |
| Jul 30, 2017 at 23:01 | comment | added | Deane Yang | Given a $2$-by-$2$ matrix $A$, could you write down $4$ matrices that would be an orthonormal basis of $T_AGL(n)$? | |
| Jul 30, 2017 at 23:00 | comment | added | Ali Taghavi | @DeaneYang The columns of $A\otimes A$ determines $n^2$ independent vectors in $\mathbb{R}^{n^2} \simeq T_A GL(n,\mathbb{R})$. Right? | |
| Jul 30, 2017 at 22:47 | comment | added | Deane Yang | Given a matrix $A$, could you provide a explicit list of the $n^2$ tangent vectors that form an orthonormal basis? Just the $n=2$ case would be good enough. | |
| Jul 30, 2017 at 22:43 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Jul 30, 2017 at 22:36 | comment | added | Ali Taghavi | @DeaneYang Thank you for your attention to my question. A metric is uniquly determined if we introduce an orthonormal frame. At each point $A\in GL(n,\mathbb{R})$ , we introduce $n^2$ independent tangent vectors to $GL(n, \mathbb{R})$ at $A$.Theses $n^2$ vectors are the columns of the matrix $A\otimes A$ for the first metric and $A\otimes A^{tr}$ for the second meyric. Here "tr" is "transpos". | |
| Jul 30, 2017 at 22:27 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Jul 30, 2017 at 22:20 | comment | added | Deane Yang | Could you (or someone else) provide a more detailed explanation of what you mean by #1? I also don't understand #2, but I'm assuming that your explanation of #1 will also clarify #2. | |
| Jul 30, 2017 at 21:06 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Jul 30, 2017 at 16:55 | history | asked | Ali Taghavi | CC BY-SA 3.0 |