Timeline for Linear transformation that preserves the determinant
Current License: CC BY-SA 4.0
12 events
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| Jan 20 at 19:10 | history | edited | LSpice | CC BY-SA 4.0 |
Link to answer, and tidying, while this is on the front page
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| Apr 29, 2020 at 13:04 | comment | added | loup blanc | I just read your good post (+1). Your characterization of $Rank(A)$ (line 14) seems -to me- valid over $F_2$ (for any $n\times n$ matrix)) -anyway, I have no counter-example-. Where do you use your hypothesis $|K|>n$ ? | |
| Apr 26, 2020 at 13:46 | comment | added | Desiderius Severus | Do we know if this is true without the $|K|>n$ hypothesis? | |
| Oct 18, 2015 at 22:29 | comment | added | Eric Wofsey | Sure; your arguments are definitely nice and more elementary than mine. I guess the point of my previous comment is just that it is quite easy to reduce to the algebraically closed case by formal considerations (but your answer is still useful as providing a more elementary argument even in that case). | |
| Oct 18, 2015 at 22:16 | comment | added | 5th decile | Okay, but invoking the Jordan normal form (or whatever related thing) for this purpose feels a bit expensive. You seem to use it in a place where just the dimension theorem suffices. By the way, I suspect that the $|K|>n$-assumption is in fact also unnecessary. | |
| Oct 18, 2015 at 22:10 | comment | added | Eric Wofsey | For infinite fields, my argument can be adapted more simply as follows. First, my argument applies verbatim to algebraically closed fields. Now note that the identity $\det T(A)=\det(A)$ must also hold over the algebraic closure $\overline{K}$ of $K$ as well, since $M_n(K)$ is Zariski-dense in $M_n(\overline{K})$. Applying my argument in $\overline{K}$, it is easy to see that the $U$ and $V$ produced actually have entries in $K$. (In fact, because $\det(A)$ and $\det T(A)$ are polynomials of degree $\leq n$, you only need $|K|>n$.) | |
| S Oct 18, 2015 at 0:10 | history | suggested | Fan Zheng | CC BY-SA 3.0 |
changed dim to \dim
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| Oct 17, 2015 at 23:42 | review | Suggested edits | |||
| S Oct 18, 2015 at 0:10 | |||||
| Oct 17, 2015 at 22:45 | history | edited | 5th decile | CC BY-SA 3.0 |
added 37 characters in body
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| Oct 17, 2015 at 21:09 | history | edited | 5th decile | CC BY-SA 3.0 |
deleted 6 characters in body
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| Oct 17, 2015 at 21:00 | history | edited | 5th decile | CC BY-SA 3.0 |
edited body
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| Oct 17, 2015 at 20:55 | history | answered | 5th decile | CC BY-SA 3.0 |