# Michael spivak

1. Mar 27, 2010

### jem05

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