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Rotational invariance

  1. Apr 23, 2010 #1
    suppose that X and Y are independent and each rotationally invariant on Rk

    a) Let P denote any orthogonal projection with dim P = k1
    determine the distribution of the correlation coefficient r= X'PY/(|PX||PY|)

    I think r is a special case of ∑(Xi-barX)(Yi-barY) = X'PX where P = I-n^(-1)11'
    but what should I do after?



    b) Let X = X1 with Xi on Rki where i =1,2 and prove
    X2
    - Each Xi is rotationally invariant in its own right

    I tried to prove Maxwell-Hershell [Maxwell-Hershell: x1,...,xn iid N(0,σ2) iff x is rotationally invariant and x1,..,xn are independent] but i was not sure...

    - If Vi is in R[from K1 to K] with V'V = I, then V'X = X1

    This one, I was not sure how to start T-T
     
    Last edited: Apr 23, 2010
  2. jcsd
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