Proof That A & A^T Have the Same Nullspace (Kernel)

[stanley.st]

Hello,

can you help me with the proof? If A is normal
$$A^TA=AA^T$$
then A and A^T have the same nullspace (kernel). And
$$||Ax||=||A^Tx||$$

Thank you.

 

[trambolin]

$A^TAx = 0 = AA^Tx$ for the vectors in the nullspace right?

 

[stanley.st]

don't understand your hint.

 

[trambolin]

multiply both by $x^T$ from the left, what is the resulting expression in terms of norms?

 

[stanley.st]

yes, I understand, thank you