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Topic: Eigenvalues (Read 545 times) |
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ThudnBlunder
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Given a 2n x 2n matrix whose entries are all odd integers, show that its eigenvalues cannot be odd integers.
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Sameer
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Re: Eigenvalues
« Reply #1 on: Dec 5th, 2006, 9:58pm » |
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Consider matrix
A = a1,1 a1,2 ... a1,2n
a2,1 a2,2 ... a2,2n
...
a2n,1 a2n,2 ... a2n,2n
Then eigenvector can be defines as L such that
AX=LX
So (a1,1+a2,1+...a2n,1) = L1
(a1,2+a2,2+...a2n,2) = L2
...
(a1,2n+a2,2n+....+a2n,2n) = L2n
Since all the elements are odd summation of 2n odd numbers is also even. Thus all L1,L2...L2n is even.
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| « Last Edit: Dec 5th, 2006, 9:59pm by Sameer » |
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towr
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Re: Eigenvalues
« Reply #2 on: Dec 6th, 2006, 1:21am » |
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If I had to hazard a guess
I'd say A2 has all even integers, so it's eigenvalues are all even, and they're the squares of the eigenvalues of A, which can thus not be odd.
On the other hand, that's assumming the eigenvectors aren't all even. But then, if they were you could divide them by the highest common factor of two. So errr, I'm left to hope there isn't a (reduced) eigenvector with an even number of odd numbers..
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| « Last Edit: Dec 6th, 2006, 1:22am by towr » |
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Eigenray
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Re: Eigenvalues
« Reply #3 on: Dec 6th, 2006, 6:11am » |
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on Dec 6th, 2006, 1:21am, towr wrote:| So errr, I'm left to hope there isn't a (reduced) eigenvector with an even number of odd numbers.. |
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The number of odd numbers can be even all it wants, it just can't be 0.
My solution: Suppose A has an odd integer eigenvalue L. Then Ax=Lx for some rational vector x; multiplying by an integer, and dividing by a power of 2 if necessary, we may assume x is integral, with at least one odd coordinate. Now, Ax=Lx means that, mod 2,
xi = Lxi = [sum] aijxj = [sum] xj
for all i. Since the RHS is independent of i, all the xi must be odd. But then [sum] xj is even, a contradiction.
(On the other hand, an odd array of odd size can have odd eigenvalues, but they can't all be odd integers, since the determinant is even, except in the 1x1 case.)
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towr
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Re: Eigenvalues
« Reply #4 on: Dec 6th, 2006, 6:30am » |
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on Dec 6th, 2006, 6:11am, Eigenray wrote:| The number of odd numbers can be even all it wants, it just can't be 0. |
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Sometimes I look back at something I wrote, and wonder what I was thinking..
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flamingdragon
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Re: Eigenvalues
« Reply #5 on: Dec 6th, 2006, 3:35pm » |
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LOL
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