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Homework Help: Similar matrices

  1. Jun 6, 2010 #1
    1. The problem statement, all variables and given/known data

    Let F(x)=(x-1)(x-2)(x-3)(x-4) be the charecteristic polynomial of A. Find the minimal polynomial of A^3

    2. Relevant equations



    3. The attempt at a solution
    A is similar to a diagnol matrix with 1,2,3,4 on the diagnol, lets call it B.
    We know that A is diagnizable because The degree of the charecteristic polynomial of a matrix is always equal to its size, and if a matrix of size n has n eigenvalues then it si diagnizable
    So [tex] A=P^{-1}BP [/tex]
    [tex] A^2=P^{-1}BPP^{-1}BP=P^{-1}B^2P [/tex]
    [tex] A^3=P^{-1}B^2PP^{-1}BP=P^{-1}B^3P [/tex]

    And so B^3 is a diagnizable amtrix with 1, 8, 27, 64 on its diagnol, which means that its charecteristic polynomial is g(x)=(1-x)(8-x)(27-x)(64-x) and because each value in the charecteristic polynomial must apear in the minimal polynomial g(x) si also the minimal polynomial of A^3.
    Is that correct?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 6, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    that sounds reasonable to me, to see it another way, assume u is an eiegnevector of A with e-val lambda, and consider the action of A^3
    [tex] A^3u = AAAu = AA\lambda u= A\lambda^2 u= \lambda^3 u[/tex]

    so lambda^3 is an eigenvalue of A^3, wth eignevector u
     
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