I am having difficulties to solve the following question. Would you please help me?
suppose that X and Y are independent and each rotationally invariant on R^k
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 R^k1 where i =1,2 and prove
- Each Xi is rotationally invariant in its own right
- If Vi is in R[from K1 to K] with V'V = I, then V'X = X1
I tried so hard to solve those two but i got stuck. Please help me