If $AA^{T}B=AA^{T}C$, then $A^{T}B=A^{T}C$

Question

Is it true that for any suitable matrices $B, C$ if If $AA^{T}B=AA^{T}C$, then $A^{T}B=A^{T}C$?

I think yes, but my reasoning seems sloppy. In particular, I think if a vector belongs to the null space of $AA^{T}$, then it also belongs to the null space of $A^{T}$. Is this claim true. If the claim is true, the question is trivial. I think the claim is true because the dimensions of both $A$, $AA^{T}$ are same. Am I right? Any hints? Thanks beforehand.

Answer 1

If $A$ has an inverse $A^{-1}$ then we have

$$AA^T B = AA^T C$$ $$A^{-1} (A A^T B) = A^{-1} (A A^T C)$$

$$(A^{-1} A) A^T B = {(A^{-1} A) A^T C}$$

$$A^T B = A^T C$$

making use of the fact that matrix multiplication is associative.

I’ll leave it to someone else to answer your real question.

Answer 2

The answer to the problem is true if the matrix $A$ is a real matrix, that is, consists of only real entries. This follows from the positive semi-definiteness of $AA^{T}$. We have $$AA^{T}x=0$$ $$\implies x^{T}AA^{T}x=0$$ $$\implies (A^{T}x)^{T}(A^{T}x)=0$$, which, owing to the real nature of matrices, is true iff $A^{T}x=0$. Of course, we assume that $x$ is a real vector.

Answer 3

I assume that $A$ is a real matrix. Let $A^+$ be the pseudo inverse to $A$. Then: $$ AA^TB=AA^TC\implies A^+AA^TB=A^+AA^TC. $$ But $A^+AA^T=A^T$ (see here, under "Identities") so the claim follows.