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Michael spivak

  1. Mar 27, 2010 #1
    1. The problem statement, all variables and given/known data
    im trying to solve in spivak's comprehensive intro to smooth manifolds, p.103 num. 31
    it's a pretty long question but i am stuck at 1 specific part.
    i have a matrix A in GL(n,R) and i showed that it can be written uniquely as A = A1.A2
    where A1 in O(n) (ie A.A^t = I) and A2 is definite positive (ie <tv,v> >0 where t: R^n --> R^n self adjoint and A would be the representation matrix.
    now prove that A1 and A2 are continuous functions of A.

    2. Relevant equations

    i proved that A1 = (A^t)^-1 . B where B^2 = A^t.A and i proved that A2=B
    moreover i have B diagonalizable and positive definite.

    3. The attempt at a solution

    i proved A^t , and A^-1 , and B^2 cont functions of A. i still need B a cont function of A.
    So really, my problem is proving the square root a cont. function of A.
    there is a hint in the book which i did not use, i donno if it helps,
    if A^(n) --> A and A^(n) = A1^(n).A2^(n) then some subsequence of A1^(n) converges
  2. jcsd
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