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Matrix Diagonalization

  1. Mar 21, 2009 #1
    1. The problem statement, all variables and given/known data
    My linear algebra textbook defines...

    similar matrices: A = C^-1BC
    diagonalized similar matrices: A = CDC^-1
    A^n = C^-1*D^n*C

    Why do the C^-1 and C's get switched around between the definitions? Doesn't order of multiplication matter? Are these the correct definitions? Is A^n really the opposite of the definition for diagonalized similar matrices, or is this an error?


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 21, 2009 #2

    Hurkyl

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    That couldn't possibly have been the definition your book gave. What did you omit?
     
  4. Mar 21, 2009 #3
    Nope, that's it! He has us using his own lecture notes, instead of a published textbook. This is why I thought it could potentially be an error.

    So if I'm diagonalizing a matrix, there exists C such that A = CDC^-1 but when I want to find A^n, I solve C^-1*D^n*C (the C inverse and C switch spots). Is that on purpose?
     
  5. Mar 21, 2009 #4

    Hurkyl

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    Nothing else, really? Nothing like
    A and B are similar matrices if ...​
    or
    ... if there exists an invertible matrix C such that ...​
    ?




    P.S. if [itex]A = C D C^{-1}[/itex], then we do indeed have [itex]A^n = C D^n C^{-1}[/itex]. (For nonnegative integers n. Negative is allowed if D is invertible) And, of course, if we have the equation [itex]A = C^{-1} D C[/itex] then we can infer [itex]A^n = C^{-1} D^n C[/itex]
     
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