Prove that $\langle AA^t v, v\rangle = \langle A^t v, A^t v\rangle $

Question

Can someone help me prove that $\langle AA^t v, v\rangle = \langle A^t v, A^t v\rangle $? Thanks a lot!

PS: I asked a question fairly similar to this one before, but it in it I proved something using the above result.

Answer 1

For vectors $x$ and $y$ and a matrix $A$, we have $$ \langle Ax,y\rangle=(Ax)^ty=(x^tA^t)y=x^t(A^ty)=\langle x,A^ty\rangle $$ Now set $x=A^tv$ an $y=v$.

Answer 2

Using the fact that the inner product is defined by $\langle u,v\rangle = u^t v$, one has $$\langle AA^t v,v\rangle=(AA^tv)^t v=v^t(A^t)^tA^t v=(A^tv)^t(A^tv)=\langle A^tv,A^tv \rangle. $$