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Proving a matrix is orthogonal.

  1. Apr 29, 2012 #1
    1. The problem statement, all variables and given/known data

    Question 10a of the attached paper.


    2. Relevant equations



    3. The attempt at a solution

    If a matrix is orthogonal, its transpose is its inverse.

    The inverse [itex]U^{-1}[/itex] is defined by Ʃ[itex]U^{-1}[/itex]ij Vj = uj

    I don't know how to go about proving this. Thanks for any help!
     

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  3. Apr 29, 2012 #2

    HallsofIvy

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    There is no attached paper! Also your given definition of "inverse" can't be true because it makes no sense- you haven't said what Vj and uj are (and you must mean ui, not uj because you should be summing over j). If you are given a specific matrix, you need to answer three questions:
    1) What is its transpose?
    2) What is its inverse?
    3) Are they the same?
     
  4. Apr 29, 2012 #3
    I attached the paper about a minute after I posted; I think it's there now. :-)
     
  5. Apr 29, 2012 #4
    Also, the sum is from j=1 to n. So the ith element of the vector u is the sum of the elements of one row of the matrix U with the elements of the vector j.
     
  6. Apr 29, 2012 #5

    Office_Shredder

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    Take the inner product of vi and vj using their expansion in terms of u's, and consider how your answer relates to matrix multiplication
     
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