Whether range of matrix $A^TA$ is the same as that of $A^T$

Question

I cannot ensure whether column space of $A^TA$ is the same as $A^T$, if it is, how to prove?

Martin Argerami · Accepted Answer · 2023-11-28T08:59:47.480

2

The key observation is that $N (B)=N (B^TB) $. Then $$R (A^T)=N (A)^\perp=N (A^TA)^\perp=R(A^TA). $$

edited Nov 28 '23 at 08:59

answered Dec 17 '17 at 03:37

Martin Argerami

but why is $N(B) = N(BB^T)$? – Cedric Martens Nov 28 '23 at 04:21
It's a typo. It should have been $N (B)=N (B^TB) $, which is just $$0=Bx \iff 0=(Bx)^TBx.$$ – Martin Argerami Nov 28 '23 at 09:01

1 Answers1