Let $M(x) \in \mathbb{R}^{d\times d}$ be a matrix-valued function on $\mathbb{R}^d$ and let $F(x) = M(x)x$.
Suppose $M(x)$ is differentiable at $x$ and the rank of $M$ is constant for an openset $\Omega$ where $x \in \Omega$.
Q: I want to know the conditions under which $$ col(\nabla F(x)) = col(M(x)) $$ is true.
My attempt: Since $$ \nabla F(x) = M(x) + \textbf{D}M(x) \cdot x, $$ where $\textbf{D}M(x)$ is the Frechet derivative, it suffices to figure out when $$ col(\textbf{D}M(x) \cdot x ) \subset col(M(x)) $$ is true. But I am not sure how to go from here.
Any help or suggestions would be very appreciated.