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wu :: forums    riddles    putnam exam (pure math) (Moderators: SMQ, william wu, Icarus, Eigenray, Grimbal, towr)    Find the characteristic polynomial « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: Find the characteristic polynomial  (Read 1099 times) ecoist Senior Riddler     Gender: Posts: 405 Find the characteristic polynomial   « on: Feb 22nd, 2008, 6:45pm » Quote Modify 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. IP Logged Obob Senior Riddler     Gender: Posts: 489 Re: Find the characteristic polynomial   « Reply #1 on: Feb 22nd, 2008, 7:48pm » Quote Modify 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). IP Logged Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Find the characteristic polynomial   « Reply #2 on: Feb 22nd, 2008, 8:49pm » Quote Modify Look at A2. IP Logged Icarus wu::riddles Moderator Uberpuzzler Boldly going where even angels fear to tread.     Gender: Posts: 4863 Re: Find the characteristic polynomial   « Reply #3 on: Feb 22nd, 2008, 9:44pm » Quote Modify 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. IP Logged "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous Eigenray wu::riddles Moderator Uberpuzzler     Gender: Posts: 1948 Re: Find the characteristic polynomial   « Reply #4 on: Feb 22nd, 2008, 10:10pm » Quote Modify ...and then look at tr(A). IP Logged Icarus wu::riddles Moderator Uberpuzzler Boldly going where even angels fear to tread.     Gender: Posts: 4863 Re: Find the characteristic polynomial   « Reply #5 on: Feb 23rd, 2008, 7:24am » Quote Modify Okay, even I can see it now.   IP Logged "Pi goes on and on and on ... And e is just as cursed. I wonder: Which is larger When their digits are reversed? " - Anonymous Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy   - medium   - hard   - what am i   - what happened   - microsoft   - cs => putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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