Do A and A^T have the same nullspace if A is normal?

stanley.st · Aug 10, 2010

Hello,

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

Thank you.

trambolin · Aug 10, 2010

[itex]A^TAx = 0 = AA^Tx[/itex] for the vectors in the nullspace right?

stanley.st · Aug 10, 2010

don't understand your hint.

trambolin · Aug 10, 2010

multiply both by [itex]x^T[/itex] from the left, what is the resulting expression in terms of norms?

stanley.st · Aug 11, 2010

yes, I understand, thank you

Do A and A^T have the same nullspace if A is normal?

Discussion

Similar threads

Do Eigenvalues of A and A^T have the same Eigenvectors?

Graduate A'A and A have the same null space

Graduate Given S, T, prove that ST and TS have the same eigenvalues.

Undergrad Do A/<b(x)> and A/<c(x)> have the same number of elements?

Graduate Why Do O(V) and SO(V) Have the Same Lie Algebra?

(LINALG) : Nullspace of transpose : N(A^T)