I have a smooth function $f$ from $\mathcal{R}^N$ to $\mathcal{R}^N$. For each $x\in \mathcal{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as $$ J_f(x)=S(x)\Lambda(x) {S(x)}^{-1}, $$ where the diagonal matrix valued function $\Lambda(x)$ contains the $N$ different eigenvalues of $J_f(x)$. If, given a diagonal matrix $D$, we define the function $$ G(x)=S(x)D\Lambda(x){S(x)}^{-1}, $$ can something be said about its integrability? Is there a function $g$ from some $U\subseteq \mathcal{R}^N$ to some $V\subseteq\mathcal{R}^N$ such that for all $x\in U$ we have $J_g(x)=G(x)$?
Any help is kindly appreciated!