Equality ker$(X{X^T})$=ker($X^T$)

Question

I'm trying to prove this equality:

ker$(X{X^T})$=ker$(X^T)$

One direction is obvious, but I can't figure out why if v is in ker$(X{X^T})$ then it is in ker$(X^T)$... Any help? Thank you

Answer 1

If $v\in\ker(XX^T)$, then $(X.X^T).v=0$ and therefore $v^T.(XX^T).v=0$. But$$v^T.(XX^T).v=(X^T.v)^T(X^T.v)=\bigl\lVert X^T.v\bigr\rVert^2$$and therefore $X^T.v=0$. In other words, $v\in\ker X^T$.

Answer 2

The other direction folows from $\langle XX^{T}v , v \rangle=\langle X^{T}v , X^{T}v \rangle =\|X^{T}v\|^{2}$ so $XX^{T}v =0$ implies $X^{T}v=0$.