Let $X$ be a connected open set of $\mathbb{R}^2$ and $f:X\rightarrow GL_2(\mathbb{C})$ be smooth. Under what conditions can we smoothly choose a singular value decomposition, i.e. find smooth maps $U,V:X\rightarrow U(2)$ and $\Sigma$ from $X$ to the 2x2 diagonal matrices with real, positive entries such that $f(x)=U(x)\Sigma(x)\bar{V}(x)^T$.
How should one go about attempting such a question?