All matrices are real. $A$ is a matrix of size $n \times k$ with $k < n$ and has independent columns.
The function $v(x) = \|Ax\|_1$ is a norm.
What is the matrix norm induced by $v$? Is it of the form $\|F (\cdot) Q\|_1$ for some $F$ and $Q$ which are easy functions of $A$.
By easy I mean functions made of matrix sum, matrix multiplication, pseudoinverse, transpose, matrices from SVD (or other well-known decomposition) of $A$ etc.
I was trying along the lines of this post with $y = Ax$, but there is a problem. $$ \max_{x\neq 0}\frac{v(Mx)}{v(x)} =\max_{x\neq 0}\frac{\|AMx\|_1}{\|Ax\|_1} =\max_{\substack{y\neq 0 \\ y \in \text{Range}\color{red}{(A)}}}\frac{\|AMA^\dagger y\|_1}{\|y\|_1} \color{red}{\neq} \|AMA^\dagger\|_1 $$ Here $\dagger$ denotes the Moore-Penrose pseudoinverse.