What are the conditions to find $A$ such that $AB=BD$ where $D$ is block diagonal?

Question

I have two known matrices $B\in\mathbb{R}^{m\times 9}$ and $D\in\mathbb{R}^{9\times9}$ where $D$ is block diagonal with each block written as $D_i=\alpha_iI_{3\times 3}$ and $I_{3\times 3}$ is the identity matrix. How can find a matrix $A$ such that $$ AB=BD. $$ I noticed that if $DB^+B=B^+BD$ then the solution to the above equation is $A=BDB^+$. However, I want to caracterize the conditions on $B$ and $D$ such this solution is valid. I would appreciate any help.

Answer 1

This equation admits a solution if and only if $$ BDB^{(1)}B=BD $$ where $B^{(1)}$ is the generalized $1-$inverse of $B$. In this case, all solutions are given by $$ A=BDB^{(1)}+W(I-B^{(1)}B) $$ for some $W$ chosen arbitrary. The reader is referred to the book: "Generalized Inverses: Theory and Applications" by Adi Ben-Israel.