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        riddles >> putnam exam (pure math) >> Find the characteristic polynomial
        
(Message started by: ecoist on Feb 22nd, 2008, 6:45pm)
Title: Find the characteristic polynomial
Post by ecoist on Feb 22nd, 2008, 6:45pm
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.
Title: Re: Find the characteristic polynomial
Post by Obob on Feb 22nd, 2008, 7:48pm
[hide]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).[/hide]
Title: Re: Find the characteristic polynomial
Post by Eigenray on Feb 22nd, 2008, 8:49pm
Look at [hide]A2[/hide].
Title: Re: Find the characteristic polynomial
Post by Icarus on Feb 22nd, 2008, 9:44pm
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.
Title: Re: Find the characteristic polynomial
Post by Eigenray on Feb 22nd, 2008, 10:10pm
...and then look at [hide]tr(A)[/hide].
Title: Re: Find the characteristic polynomial
Post by Icarus on Feb 23rd, 2008, 7:24am
Okay, even I can see it now.  :P


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