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Topic: Find the characteristic polynomial (Read 964 times) 

ecoist
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Find the characteristic polynomial
« on: Feb 22^{nd}, 2008, 6:45pm » 
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Let A=(a_{ij}) be a 2009x2009 matrix with a_{ij}=2 if i+j=2010, and a_{ij}=1 otherwise. Find the characteristic polynomial, C_{A}(x), of A.


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Obob
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Re: Find the characteristic polynomial
« Reply #1 on: Feb 22^{nd}, 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 C_{A}(x)=(x1)^{1004}(x+1)^{1004}(x2010).


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Eigenray
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Re: Find the characteristic polynomial
« Reply #2 on: Feb 22^{nd}, 2008, 8:49pm » 
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Look at A^{2}.


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Icarus
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Re: Find the characteristic polynomial
« Reply #3 on: Feb 22^{nd}, 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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Eigenray
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Re: Find the characteristic polynomial
« Reply #4 on: Feb 22^{nd}, 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 23^{rd}, 2008, 7:24am » 
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Okay, even I can see it now.


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"Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? "  Anonymous



