Timeline for Problems concerning subspaces of $M_{n}(\mathbb{Q}) $
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jun 8, 2021 at 19:20 | vote | accept | Sky | ||
| May 17, 2021 at 17:42 | comment | added | Vladimir Dotsenko | Somehow, I am not sure that the tags are not misplaced. | |
| May 17, 2021 at 17:32 | history | edited | Sky | CC BY-SA 4.0 |
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| May 17, 2021 at 17:17 | history | edited | Sky | CC BY-SA 4.0 |
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| May 17, 2021 at 9:36 | comment | added | LSpice | @Zero, do you want $B : (x, y, z) \mapsto (y, z, 2x)$? (I'm thinking of the obvious embedding of $\mathbb Q(\sqrt[3]2)$ in $\operatorname M_3(\mathbb Q)$, but maybe you have something else in mind!) | |
| May 17, 2021 at 9:28 | answer | added | Robert Bryant | timeline score: 21 | |
| May 17, 2021 at 8:39 | comment | added | user130903 | This question does not have the same answer for $\mathbb Q$ as for $\mathbb R$. Let $A$ be the 3 x 3 diagonal matrix with entries (2,1,1) and let $B$ be the matrix of the linear map $(x,y,z)\mapsto (y,z,x)$, then $A$ and $B$ span such a space over $\mathbb Q$, since there is no third root of $2$ in $\mathbb Q$. However, $\rho(3)=1$. | |
| May 17, 2021 at 8:25 | comment | added | R.P. | @LSpice Yes, informal suggestion. However you could define it presumably by intersecting the subspace with the unit hypersphere (in $\mathbb{R}^{n^2}$) and taking the distance between the intersections. | |
| May 17, 2021 at 7:47 | comment | added | LSpice | This seems related to your earlier question mathoverflow.net/questions/392878/…. It's not a duplicate (at least I can't see how), but it might be better to figure out one question than to ask several closely related questions in succession. | |
| May 17, 2021 at 7:45 | comment | added | LSpice | @RP_, how should one define closeness of subspaces? Or is it just an informal suggestion? | |
| May 17, 2021 at 7:07 | comment | added | R.P. | The naive idea is that, since Q is dense in R, we can approximate a subspace N defined over R by a subspace N' defined over Q, and if N' is close enough to N then it should share the property of avoiding non-zero non-invertible matrices. Have you tried to make this work? | |
| May 17, 2021 at 6:07 | history | asked | Sky | CC BY-SA 4.0 |