I have a map $f:\mathbb{R}^{n\times n}\times \mathbb{R}^n\to\Bbb R^n$ that takes a matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ that is full rank and a vector $\mathbf{x}\in\mathbb{R}^n$ that is non-zero such that $f(\mathbf{A},\mathbf{x})=\mathbf{Ax}$. Let $v:=(\mathbf{A},\mathbf{x})$, I am looking to bound the following: $$\frac{\lVert v-v'\rVert_2}{\lVert f(v)-f(v')\rVert_2}\leq L,$$ by some constant, say $L$. Would I be right in thinking that I could find $L$ by bounding the smallest eigenvalue of the derivative $\nabla f(v)$? The issue I face is that the map $f$ is not square (and not injective) so I can not use the classical inverse function theorem.
Thanks in advance for your help.