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Find formulas for the entries of M^n

  1. Apr 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [itex]M = \begin {bmatrix} 8 & -1 \\ 2 & 11 \\ \end{bmatrix}[/itex]

    Find formulas for the entries of [itex]M^n[/itex], where n is a positive integer.

    2. Relevant equations
    N/A

    3. The attempt at a solution
    I honestly have no clue where to start. We recently covered diagonalization, but I can't see how this relates.
     
  2. jcsd
  3. Apr 26, 2012 #2
    Let's suppose D is the transform that diagonalizes M, and denote [itex] D M D^{-1} = \tilde{M} [/itex]. Then
    [tex] M^n = M M M ... M = D^{-1} D M D^{-1} D M D^{-1} D M ... M D^{-1} D = D^{-1} \tilde{M}^n D. [/tex]
     
  4. Apr 26, 2012 #3

    HallsofIvy

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    Also, The diagonal entries in [itex]\tilde{M}[/itex] are the eigenvalues of the matrix M and the columns of D are the corresponding eigenvectors of M.
     
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