Timeline for When matrices commute
Current License: CC BY-SA 3.0
25 events
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| Jan 21, 2021 at 22:57 | comment | added | Alex Ravsky | Related. | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Oct 20, 2016 at 16:08 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Oct 20, 2016 at 15:56 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| Oct 20, 2016 at 15:51 | comment | added | Will Jagy | en.wikipedia.org/wiki/When_Animals_Attack! | |
| Oct 20, 2016 at 13:36 | comment | added | Nate Eldredge | I don't get the joke in the title, despite having read the links (including the one to a deleted question). I am sure it is extremely clever, but for the benefit of us lesser humans who still might find the math interesting, perhaps it would be a good idea to use a title that simply states the mathematical content of the question? | |
| Nov 22, 2011 at 19:58 | vote | accept | Will Jagy | ||
| Nov 22, 2011 at 8:04 | answer | added | Pierre-Yves Gaillard | timeline score: 8 | |
| Jun 9, 2011 at 16:17 | vote | accept | Will Jagy | ||
| Nov 22, 2011 at 19:58 | |||||
| May 24, 2011 at 18:59 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| May 24, 2011 at 18:50 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| May 24, 2011 at 18:39 | history | edited | Will Jagy | CC BY-SA 3.0 |
conjecture
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| May 24, 2011 at 1:56 | answer | added | Will Jagy | timeline score: 7 | |
| May 24, 2011 at 1:24 | comment | added | Richard Stanley | The general theory of which matrices $A$ commute with a given matrix $M$ is covered in Section 1.10.1 of math.mit.edu/~rstan/ec/ec1.pdf. This treatment focuses on counting the number of such matrices $A$ over a finite field and assumes that $A$ is invertible, but the classification carries over readily to any field, with or without the invertibility assumption. See page 108 for some references. | |
| May 23, 2011 at 20:59 | comment | added | Will Jagy | Tommaso, I think the cuboid animals are in $$ $$ mathoverflow.net/questions/65372/random-polycube-shapes $$ $$ while the phrase "I am shocked, I tell you, shocked" comes from the review of the television show "When animals attack" and, originally, from the movie Casablanca, imdb.com/title/tt0034583 and Rick: How can you close me up? On what grounds? Captain Renault: I'm shocked, shocked to find that gambling is going on in here! [a croupier hands Renault a pile of money] Croupier: Your winnings, sir. | |
| May 23, 2011 at 20:50 | comment | added | Tommaso Centeleghe | Am I missing something not understanding what are the cuboids animals? | |
| May 23, 2011 at 20:44 | comment | added | Gerhard Paseman | What? No cuboid animals? Or even hints of one? I am shocked, I tell you, shocked! Gerhard "We've Got Trouble, Right Here" Paseman, 2011.05.23 | |
| May 23, 2011 at 20:43 | comment | added | David E Speyer |
Fair enough, but the example will get messier then. Let $B = \begin{pmatrix} 0 & 1 \\ 2 & 0 \end{pmatrix}$, so $B$ has eigenvalues $\pm \sqrt{2}$. Let $A = 3+2 B$. Then the eigenvalues of $A$ are $3 \pm 2 \sqrt{2}$, so $\det(A) = (3+2\sqrt{2})(3-2\sqrt{2})=1$. Then $C$ commutes with $A$ if and only if it commutes with $B$. The integral matrices commuting with $C$ are $\mathbb{Z}[B]$, whereas $\mathbb{Z}[C] = \mathbb{Z}[2B]$.
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| May 23, 2011 at 20:35 | history | edited | Will Jagy | CC BY-SA 3.0 |
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| May 23, 2011 at 20:35 | comment | added | Denis Serre | David, this $A$ is not $SL_2$. | |
| May 23, 2011 at 20:31 | comment | added | David E Speyer |
Even then, though, you can't hope for integrality. For example, look at matrices commuting with $A=\begin{pmatrix} 2 & 0 \\ 0 & 4 \end{pmatrix}$. These are precisely the diagonal matrices. But $\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$ is an integer polynomial in $A$ iff $a \equiv b \mod 2$.
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| May 23, 2011 at 20:28 | comment | added | David E Speyer | There are some true statements that are like this. If the characterstic polynomial of $A$ has distinct roots, and if you are working over a field $K$, then all matrices that commute with $A$ are polynomials in $A$. See the first part of my answer here mathoverflow.net/questions/55620/commuting-matrices-in-gln-z | |
| May 23, 2011 at 20:25 | comment | added | Tommaso Centeleghe | I think you at least want A to have eigenvalues with multiplicity one. Otherwise it is definitely not true. Right? | |
| May 23, 2011 at 20:24 | comment | added | David E Speyer | No. Take $A$ to be the identity. | |
| May 23, 2011 at 20:20 | history | asked | Will Jagy | CC BY-SA 3.0 |