What kind of matrix $B$ can preserve row space of $A$ when their product, like $BA$

Question

Ok, I am not very good at linear algebra. But When I read a post:

which someone try to prove it by using Null space of $A^T A$ and $A$. An idea pops up. For any matrix $A$ with size of $m\times n$, what different between row space of $BA$ and $A$, and more importantly, what kind of $B$ can make sure the row space of $A$ and $BA$ are the same?

Well, if $B$ is an full rank square matrix, no doubt that row space would not change. But the post above shows that when $B=A^T$ the row space would not change either. And here $B$ does not have to be square.

So, can anyone tell me what kind of $B$ will preserve row space of $A$ when $A$ times $B$?

Thinks, I should say that all matrix are real.

If you have read the comment in your attachment, then the statement is not true for complex matrices, it is true only for real matrices. — Sarvesh Ravichandran Iyer, Oct 20 '16 at 09:42
The problem would have everything to do with the intersection of the null space of $B$ and the range (columns space) of $A$, would it not? Think about it … — Harald Hanche-Olsen, Oct 20 '16 at 11:17

score 1 · Answer 1 · answered Oct 21 '16 at 01:26

Thinks for Harald Hanche-Olsen's hint, I figured it out.

Because null space of $BA$ always included in null space of $A$. For preserve row space of $A$, just need to make sure those two null space equal.

$Ax$ are column combinations of $A$, if $A$'s column space dose not contain any vector which perpendicular to B, OR like Harald Hanche-Olsen said, the null space of $B$ and column space of $A$ only intersected at zero vector, then we got it.

Thinks for Harald Hanche-Olsen again. Very good hint.

What kind of matrix $B$ can preserve row space of $A$ when their product, like $BA$

1 Answers1