Timeline for An $n \times n$ matrix $A$ is similar to its transpose $A^{\top}$: elementary proof?
Current License: CC BY-SA 3.0
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| May 30, 2018 at 22:24 | comment | added | Sungjin Kim | After many years this question was first asked, I cannot find anything meeting all of my requirement. Actually, finding Smith Normal Form can be done through elementary row/column operations. So, it is already elementary enough, and I do not have any reason left for avoiding such a simple and powerful tool. Also, SNF provides not only the existence of $P$, but also an explicit way to write down a such $P$. | |
| S Oct 9, 2017 at 12:07 | history | edited | R.P. | CC BY-SA 3.0 |
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| S Oct 9, 2017 at 12:07 | history | suggested | Rodrigo de Azevedo | CC BY-SA 3.0 |
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| Oct 9, 2017 at 11:57 | review | Suggested edits | |||
| S Oct 9, 2017 at 12:07 | |||||
| May 13, 2016 at 7:01 | comment | added | Duchamp Gérard H. E. | @user46896 The method is interesting, but not very far from Jordan canonical form (see questions V.B.1. and V.B.2.). One dreams about an algorithm providing $P$ from scratch ... | |
| S Mar 30, 2014 at 15:25 | history | bounty ended | CommunityBot | ||
| S Mar 30, 2014 at 15:25 | history | notice removed | CommunityBot | ||
| Mar 30, 2014 at 12:57 | comment | added | user46896 | @i707107 So, I hope you can work with this : sujets-de-concours.net/sujets/centrale/2003/tsi/math2.pdf , look at Partie V, it's an elementary approach. The solution here : pomux.free.fr/corriges-2003/index.html | |
| Mar 30, 2014 at 8:30 | comment | added | Sungjin Kim | @Julien I know very little french, but there is google translate. | |
| Mar 29, 2014 at 1:43 | comment | added | user46896 | @i707107 Can you read some french papers ? | |
| Mar 26, 2014 at 1:02 | comment | added | user62675 | See projecteuclid.org/euclid.pjm/1103039127. Maybe math.stackexchange.com/questions/62497/… is related. | |
| Mar 25, 2014 at 4:53 | comment | added | Sungjin Kim | @BenjaminDickman The first link shows that the solution space of $XA=A^T X$, $X=X^T$ has at least dimension $n$. However, the paper does not discuss how to obtain a "nonsingular" matrix in the solution space. That was actually a motivation of my question. | |
| Mar 24, 2014 at 23:05 | comment | added | Benjamin Dickman | Is the proof given here msp.org/pjm/1959/9-3/pjm-v9-n3-p25-p.pdf sufficiently "elementary"? Incidentally, note the remark on p.895 (pdf 4/7) comes from M. Newman; there was an earlier MO question answered using Smith's work (which you wish to avoid here...) that turned out to have a more elementary solution from M. Newman (mathoverflow.net/questions/151166/…). If the linked proof here does not suffice, perhaps Newman is the fellow to look to... | |
| Mar 22, 2014 at 14:25 | comment | added | Geoff Robinson | @MarkSapir : Yes, of course the fact that $A$ and $A^{T}$ have the same minimum polynomial is obvious: it was the use of rational canonical form ( which is really more general that Jordan Normal Form) that seemed to be outside the spirit of the question. | |
| S Mar 22, 2014 at 14:19 | history | bounty started | CommunityBot | ||
| S Mar 22, 2014 at 14:19 | history | notice added | user46896 | Draw attention | |
| Mar 26, 2013 at 21:10 | comment | added | Bob Terrell | This is slightly related to my question <a href="http:mathoverflow.net/questions/42072</a> | |
| Feb 22, 2013 at 3:51 | comment | added | Sungjin Kim | @Tom: This link projecteuclid.org/… | |
| Feb 20, 2013 at 4:39 | comment | added | user6976 | @Geoff: the fact that $A$ and $A^T$ have the same minimal polynomial is obvious: if $p(A)=0$, then $p(A^T)=p(A)^T=0$. | |
| Feb 19, 2013 at 23:21 | comment | added | Tom Goodwillie | In the two by two case $P$ can always be chosen to be symmetric. Is this true in general? | |
| Feb 19, 2013 at 23:19 | comment | added | Tom Goodwillie | It's easy to prove using normal forms, but you can reframe it as the statement that if $A:V\to V$ is linear then there is a nondegenerate bilinear form on $V$ such that $\langle Av,w\rangle=\langle v, Aw\rangle$ identically. Is there any chance of a proof of this that is quite different from the normal-form proofs? | |
| Feb 19, 2013 at 21:40 | comment | added | Geoff Robinson | There is a proof using rational canonical form, which basically just needs the fact that $A$ and $A^{t}$ have the same minimum polynomial, but I don't think that will be considered explicit .enough | |
| Feb 19, 2013 at 21:08 | comment | added | Sungjin Kim | @Mariano: Yeah, it might not be possible to avoid those to prove it. The matrix P coming from Jordan canonical form proof is already complicated enough. Maybe requiring to provide explicit P is not possible without those. The generalized eigenvectors are involved and an inversion that transforms Jordan block to its transpose. | |
| Feb 19, 2013 at 19:58 | comment | added | Mariano Suárez-Álvarez | This is a lot like looking for a proof which does not need the letter a in order to be written down :-) What exactly do you mean by elementary? The Jordan canonical form is basic linear algebra, really. | |
| Feb 19, 2013 at 19:40 | history | asked | Sungjin Kim | CC BY-SA 3.0 |