Given matrices $A$ and $B$ find $B'$ such that $AB = AB'$ yet $\|{B}'\|<\|B\|$.

Question

I'm working on a task where I have matrices $A$ and $B$ respectively. I need to find a matrix $B'$ such that $AB = AB'$ yet $\|{B}'\|<\|B\|$. This is mainly because currently $\|{B}\|$ is very large and which hurts the task I'm solving.

Anyone has any idea on how to solve this?

Thanks!

Answer 1

Given $A,B$, the optimization problem $$ \begin{split} \min\; & \Vert B'\Vert^2\\ \text{s.t. }\, & AB'=AB \end{split} $$ is a convex optimization problem with linear constraints and can be solved via standard techniques (whenever it has a solution).