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 [SOLVED] Matrix of order p

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Bruno
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PostSubject: [SOLVED] Matrix of order p   Sat Nov 21, 2009 12:24 am

This one is nice; I was planning on using it in Mohammad's linear algebra problem but I never saw the light at the end of the tunnel. So here it is, for itself and by itself.

Let p be a prime number, and A an (n x n) complex matrix. Suppose that A has order p and zero trace. Show that p divides n.


Last edited by Bruno on Sun Nov 22, 2009 7:47 pm; edited 1 time in total
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Mohammad
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PostSubject: Re: [SOLVED] Matrix of order p   Sun Nov 22, 2009 12:10 am



Last edited by Mohammad on Sun Nov 22, 2009 12:09 pm; edited 3 times in total
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Bruno
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PostSubject: Re: [SOLVED] Matrix of order p   Sun Nov 22, 2009 12:43 am

Beautiful solution! cheers

My solution begins the same way as yours but then uses the fact that (x^p-1)/(x-1) is irreducible over the rationals, so its roots are linearly independent over the rationals. Writing A in Jordan form you see that all the roots of x^p-1 must appear as many times on the diagonal, so n = p x q for some integer q.

I like your solution better though. Less machinery!
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Mohammad
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PostSubject: Re: [SOLVED] Matrix of order p   Sun Nov 22, 2009 11:55 am

I am going to prove the third root of 2 is irrational. Suppose it is rational so for some natural numbers p and q we have $q^3+q^3=p^3$ now by last theorem of Fermat we reach to a contradiction. I got this sharp critic in contest preparation class when I attacked a tricky problem in number theory by Chevalley theorem, I was 20 and felt totally embarrassed among other students Embarassed So I don't dare give a solution if it can be solved using famous theorem unless I have to bounce . I don't mean any offense it was just an experience to be said. Very Happy
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Mohammad
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PostSubject: Re: [SOLVED] Matrix of order p   Sun Nov 22, 2009 12:26 pm

One more thing regarding to this problem! I am not sure but maybe your proof also suggests that these eigenvalues form a group under multiplication ha? bounce
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peyman
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PostSubject: Re: [SOLVED] Matrix of order p   Sun Nov 22, 2009 10:21 pm

nice solution! I almost had it, except I didn't know the multinomial expansion so I got stuck not knowing what to do with the mixed products of roots of unity. I'm really enjoying the problems and their solutions so keep it up!
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PostSubject: Re: [SOLVED] Matrix of order p   Sun Nov 22, 2009 10:23 pm

I'm not sure if you noticed, but I'm very old now, so treat me with respect What a Face Very Happy !
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Bruno
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PostSubject: Re: [SOLVED] Matrix of order p   Mon Nov 23, 2009 2:56 am

peyman wrote:
I'm not sure if you noticed, but I'm very old now, so treat me with respect What a Face Very Happy !

Of course we treat you with respect no matter your age, wise old man Wink
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