Let $X \in \mathbb{R}^n$ be a random vector with distribution $F_X$.
Now let $A$ be an $m \times n$ matrix and define \begin{align} Z= AX, \end{align} with distribution $Z\sim F_Z$.
My question:
Can we show sufficient and necessary conditions on matrix $A$ under which $F_X$ is unique for every $F_Z$.
Clear, it is sufficient that $A$ is left invertible, that is there exists a matrix $C$ such that $CA=I$. Is this also, necessary?