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Linear Algebra - Find Orthogonal Matrix Q that diagonals

  1. Apr 24, 2016 #1
    1. The problem statement, all variables and given/known data
    I'm told to find the matrix Q of the matrix A
    equation 1.PNG

    2. Relevant equations


    3. The attempt at a solution

    So my problem is that in the answer key they have S = (1/3)... and I have no idea where this 1/3 comes from. I get an equivalent answer for X_1, X_2, and X_3
    S = [X_1, X_2, X_3] but when I form this matrix with the values for X_1, X_2, X_3 I'm not able to factor out a 1/3. Does anyone know where this value comes from?
    This is the solutions

    equation2.PNG
    and this is my attempt, again I would get the same answer but have no idea were the 1/3 came from.
    IMG_20160424_204918869.jpg
    thanks for any help
     
  2. jcsd
  3. Apr 24, 2016 #2

    blue_leaf77

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    An orthogonal matrix is a unitary matrix. Therefore ##Q^*Q = QQ^*=I##. This is why the vectors making up the column of ##Q## must be orthonormal - 1/3 is a normalization constant.
     
  4. Apr 24, 2016 #3
    How do I calculate this normalization constant? Thanks for your help. Like I'm exactly sure which vector I'm normalizing and getting 1/3. But I know to normalize a vector it's

    V/||V||

    Thanks for your help!
     
  5. Apr 24, 2016 #4

    blue_leaf77

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    A vector is said to be normalized if its norm equal unity.
     
  6. Apr 24, 2016 #5
    So it's just

    1/sqrt((1+1+1)^2)?
     
  7. Apr 24, 2016 #6

    blue_leaf77

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    I don't understand what you want to say there.
    Suppose you have three component vectors ##u = (a,b,c)^T##. In order to make it normalized you have to add a common constant by yourself to the vector: ##u = \gamma (a,b,c)^T##. By requiring its norm to be equal to one, namely ##\gamma^2(|a|^2 + |b|^2 + |c|^2) = 1##. This way you can find ##\gamma## in terms of the components of ##u##. For a general unitary matrix, ##\gamma## can be complex but since here the problem asks for an orthogonal matrix, ##\gamma## must be real.
     
  8. Apr 24, 2016 #7
    Oh I see that makes more since to me know. So it's for the whole Matrix S? When I square all the terms in S and sum each row individually I get 9, so the constant must be 1/sqrt(9) = 1/3.

    What if each row doesn't sum up to the same value like it did in this case?

    Thanks for your help!
     
  9. Apr 24, 2016 #8

    blue_leaf77

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    Then you cannot pull the common normalization constant out of the matrix's bracket. Nothing wrong with that.
     
  10. Apr 24, 2016 #9
    ah ok thanks for your help
     
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