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General truth about elements of a matrix to the n-th power?

  1. Jan 19, 2012 #1
    Say we have a matrix P with eigenvalues [itex]\lambda_1, \cdots, \lambda_n[/itex] (possibly some are the same) and P can be diagonalized, then we can always say that the element on the a'th row and b'th column of P^n is equal to [itex]P^n(a,b) = \sum_{i = 1}^n \alpha_i \lambda_i^n [/itex] with [itex]\alpha_i[/itex] independent of n (but dependent on a and b).

    Correct? And I don't think the conditions can be weakened?
     
  2. jcsd
  3. Jan 19, 2012 #2

    micromass

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    Seems indeed correct.
     
  4. Jan 19, 2012 #3
    I find it to be rather pretty :)
     
  5. Jan 19, 2012 #4

    micromass

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    Indeed, it IS pretty!! :biggrin:
     
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