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Homework Help: A linear algebra problem.

  1. Nov 25, 2008 #1
    1. The problem statement, all variables and given/known data
    A is a 2x2 matrix with eigenvectors 1,-1 and 1,1 with respective eigenvalues 2 and 3. x is 2,0. Find (A^4)x


    2. Relevant equations
    I know (A^k)v=(lamda^k)v
    But I just don't know how to solve this to find A and then multiple it by x

    3. The attempt at a solution
    See above

    Thanks a lot for anyone's help or input!
     
  2. jcsd
  3. Nov 25, 2008 #2

    Vid

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    The eigenspace is two dimensional since there are two distince eigenvalues. Thus, A is diagonalizable. Then....
     
  4. Nov 25, 2008 #3

    HallsofIvy

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    More simply, <2, 0>= <1, -1>+ <1, 1>. Apply A to that. You don't need to determine A itself.
     
  5. Nov 25, 2008 #4

    Dick

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    You don't need to find A. (2,0)=(1,-1)+(1,1). How would you find A^4((1,-1))?
     
  6. Nov 25, 2008 #5
    Ah I got it, (A^k)x=(lamda1^4)v1+(lamda2^4)v2

    Thanks a lot for pointing out that those two vectors added up to x, I overlooked that. Would this be possible otherwise?
     
  7. Nov 25, 2008 #6

    Dick

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    If the only information you have about A is it's eigenvectors, and you can't express the vector as a linear combination of eigenvectors, then, no, you don't have enough information about A.
     
  8. Nov 25, 2008 #7
    Thank you very much you guys are awesome.
     
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