Solve in spivak's comprehensive intro to smooth manifolds

[jem05]

Homework Statement

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.

Homework 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.

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

 

[mighty2000]

to A1 and same for A2.



Thank you for sharing your question. It seems that you have made good progress so far in your problem. To prove that A1 and A2 are continuous functions of A, you will need to use the fact that A^t, A^-1, and B^2 are continuous functions of A. This means that the composition of these functions, which is A1, must also be continuous. Additionally, since B is diagonalizable and positive definite, it can be written as B = PDP^-1 where D is a diagonal matrix with positive entries. This means that the square root of B can be written as the square root of D, which is also a diagonal matrix with positive entries. Therefore, the square root of B is a continuous function of B, and since B is a continuous function of A, the square root of B is also a continuous function of A. This shows that A2 = B is a continuous function of A. Combining this with the fact that A1 is also a continuous function of A, we can conclude that A1 and A2 are both continuous functions of A. I hope this helps you in solving your problem. Good luck!