$Null (A) \subseteq Null(A^T A)$ if and only if $Null(A^T A) \subseteq Null(A)$

Question

Let $A$ be an $m \times n$ matrix. Show that $Null (A) \subseteq Null(A^TA)$ if and only if $Null(A^TA) \subseteq Null(A)$.

I can not really come up with the idea how to show that. Can someone help me? I would really appreciate that.

In the duplicate question, just change $A$ into $A^T$. – egreg Jun 29 '17 at 20:13 — egreg, Jun 29 '17 at 20:13

score 1 · Answer 1 · answered Jun 29 '17 at 20:12

1

$N(A) \subset N(A^*A)$ is obvious.

Conversely, if $A^*A x = 0$, then $\langle A^*Ax , x \rangle = 0 \Rightarrow \langle Ax,Ax \rangle = 0 \Rightarrow \|Ax\|^2 = 0 \Rightarrow Ax = 0$.

answered Jun 29 '17 at 20:12

Daniel Xiang

but there is a transpose of A, not multiplication of A by itself – Jun 29 '17 at 21:42
the transpose, $A^$, is defined as an operator such that $\langle Ax,y \rangle = \langle x,A^y\rangle$, where $\langle \cdot,\cdot \rangle$ is the inner product. – Daniel Xiang Jun 30 '17 at 13:14

1 Answers1