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   Find the characteristic polynomial
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   Author  Topic: Find the characteristic polynomial  (Read 964 times)
ecoist
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Find the characteristic polynomial  
« on: Feb 22nd, 2008, 6:45pm »
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Let A=(aij) be a 2009x2009 matrix with aij=2 if i+j=2010, and aij=1 otherwise.  Find the characteristic polynomial, CA(x), of A.
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Obob
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Re: Find the characteristic polynomial  
« Reply #1 on: Feb 22nd, 2008, 7:48pm »
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It is possible to write down 2009 linearly independent eigenvectors for A.  There are 1004 with eigenvalue 1, 1004 with eigenvalue -1, and 1 with eigenvalue 2010.  So CA(x)=(x-1)1004(x+1)1004(x-2010).
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Eigenray
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Re: Find the characteristic polynomial  
« Reply #2 on: Feb 22nd, 2008, 8:49pm »
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Look at A2.
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Icarus
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Re: Find the characteristic polynomial  
« Reply #3 on: Feb 22nd, 2008, 9:44pm »
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Nice, but do you have a good argument for showing that the eigenvalues of A are equally split between -1 and 1, or even that the final eigenvalue is 2010, not -2010?
 
I'm not seeing it myself (other than the long way that leads directly there), but then that doesn't say much.
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Re: Find the characteristic polynomial  
« Reply #4 on: Feb 22nd, 2008, 10:10pm »
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...and then look at tr(A).
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Icarus
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Re: Find the characteristic polynomial  
« Reply #5 on: Feb 23rd, 2008, 7:24am »
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Okay, even I can see it now.  Tongue
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