Say I have a positive-definite (in particular invertible) matrix $A \in \mathbb{R}^n \times \mathbb{R}^n$. Suppose also that $U\subseteq \mathbb{R}^n$ is a bounded domain with smooth boundary. Is the linear map $$ T:U\to T(U),\quad x\mapsto Ax $$ a diffeomorphism on $U$?
I feel like the answer should be yes but I wanted to make sure (by the way, this is not a homework problem). Could anyone confirm my hypothesis? Or am I wrong to assume this?
Thanks!