Range A = range A^tA

Question

let $A\in\mathbb{R}^{m\times n}$ with $m\geq n$. ¿Is it true that rank $A$ is equal to rank $A^tA$?. How show this?

Thanks!

What you can prove is that $\text{range }A^\top=\text{range }A^\top A$. — Ted Shifrin, Dec 24 '13 at 01:56
Thanks! It is true, I duplicated the question. I am sorry, and thanks for the link with the answer. — yemino, Dec 24 '13 at 01:58

score 1 · Answer 1 · answered Dec 24 '13 at 01:48

1

Counterexample $$A={1\choose0}\quad A^TA=(1)$$

Actually, if $m\neq n$, then range of $A$ is subspace of $\mathbb R^m$ while range of $A^TA$ is subspace of $\mathbb R^n$.

answered Dec 24 '13 at 01:48

Shuchang

sorry, it was rank, not range. – yemino Dec 24 '13 at 01:53
@user106306 Well, I think you found the answer now. – Shuchang Dec 24 '13 at 01:59
yes, thanks for your help and time. (Although, it is always good to see an alternative proof). – yemino Dec 24 '13 at 02:02

1 Answers1