Reduction of structure group of real vector bundles I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book gives a hint by asking to show that the general linear group is the direct product of the orthogonal group and the group of symmetric positive definite matrices. I proved this using the polar decomposition.
Now if I have a cocycle $ g_{\alpha\beta} $ I can write it as $ g_{\alpha\beta} = u_{\alpha\beta} p_{\alpha\beta} $ where $ u $ is orthogonal and $ p $ is symmetric positive definite. $ u $ and $ p $ vary smoothly with $ g $. What I need to do now is to define a map $ \lambda_{\alpha}:U_{\alpha}\rightarrow GL(n,R)  $ so that $ g_{\alpha\beta} = \lambda_{\alpha} u_{\alpha\beta} \lambda_{\beta}^{-1} $. I think $ \lambda $ should depend on $ p $ but I don't know how to make $ \lambda $ well defined and get the result.
Am I on the right track? Should I use another matrix decomposition? I looked at a list of decompositions and couldn't find a better match for this problem.
I'm interested in a solution that follows the hint. The book already contains a proof using a metric and partition of unity, so the other question is not a duplicate. I don't know Riemannian Geometry so please don't use it.
Thanks

What you are looking at is actually the Gram–Schmidt process, this gives the desired decomposition of a matrix $g \in GL(n)$ into $\lambda u \lambda^{-1}$, where $u \in O(n)$ in a natural way, i.e. the correspondence does not depend on choices and is smooth for a smooth family of matrices $g: U \to GL(n)$.
