If v is an eigenvector of the matrix AA^T (A multiplied by its transpose) with eigenvalue λ, show that A^Tv is an eigenvector of A^TA.
Please Help >_<
By hypothesis there exits an scalar $\displaystyle \lambda$ such that $\displaystyle (AA^t)v=\lambda v$. Then,
$\displaystyle (A^tA)(A^tv)=A^t[(AA^t)v]=A^t(\lambda v)=\lambda(A^tv)$
That is, $\displaystyle A^tv$ is an eigenvector of $\displaystyle A^tA$.
Regards.