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SqueeSpleen
 one year ago
Best ResponseYou've already chosen the best response.0I'm trying to deal with the following problem: A={{0,1},{1,1}} X^2X1 is the charasteristic polynomial of A, so: A^2AI_2=0 I had a similar problem in the theory, but it was simplier. It was: A={{0,1},{1,2}} And it charasteristic polynomial was: X^22X+1=(X1)^2 By the division algorythm we have: P(X) \in R[X] and a_n, b_n in R such that: X^2=(X1)^2.P(X)+a_n.X+b_n * Using X=1 we have: 1=0.P(X)+a_n.1+b_n then a_n+b_n=1 We derivate * and we have: n.X^(n1)=2(X1).P(X)+(X1)^2.P'(X)+a_n We take X=1 and we have: n=a_n So we have b_n=1n Then: X^n=(X1)^2.P(X)+n.X+1n So: A^n=n.A+(1n).I_2={{1n,n},{n,n+1}} But with: A={{0,1},{1,1}} I can't solve this, because P'(X) and P(X) don't share any roots, so I can't isolate a_n or b_n, the best I achieved was: ((1+5^(1/2))/2)^n=((1+5^(1/2))/2).a_n+b_n and other equations, but I can never isolate a term. I know A^n.{{0,1}} is {{f_n},{f_(n+1)}} where f_n is the nth fibonnaci's number, but I'm trying to prove it in this way :p Sorry for my bad English :p I don't know a lot of mathematical terms in this language, neither I'm good speaking it.

SqueeSpleen
 one year ago
Best ResponseYou've already chosen the best response.0If it is to difficult to read I can use some LaTeX, feel free to ask me for it.

SqueeSpleen
 one year ago
Best ResponseYou've already chosen the best response.0I also have the link to the book where it comes from (it isn't piracy :P, they are a class notes), but it's in Spanish.

SqueeSpleen
 one year ago
Best ResponseYou've already chosen the best response.0I diagonalized the matrix, and by the fact that if A=C.B.C^(1) then A^n = C.B^n.C^(1) and a nth power of a diagonal matrix is the matrix which has the autovalors at the nth power, we have: ... {{(1+5^(1/2))/2,(1+5^(1/2))/2},{1,1}}.{{((15^(1/2))/2)^n,0},{0,((1+5^(1/2))/2)^n}}.{{(1+5^(1/2))/2,(1+5^(1/2))/2},{1,1}}^(1) I suppose that my major mistake was to try to solve it in the same way instead of thinking what tools of the chapter were useful.

nitz
 one year ago
Best ResponseYou've already chosen the best response.0although i have studied matrices....but closed form....ill check

SqueeSpleen
 one year ago
Best ResponseYou've already chosen the best response.0http://en.wikipedia.org/wiki/Fibonacci_number#Matrix_form I'll close it.
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