Why can we know the best precoding direction after we do the SVD to the channel?for example
$\mathbf y=\mathbf H\mathbf s+\mathbf n$,now we do the SVD to the $\mathbf H$
If the $\sigma _1$ is the biggest singular value,after we normalize $V_1$,why can we say the $V_1$ is the best precoding direction?Is there any theorem can prove this?
First you have to know that the vector in $U$ and the vector in $V$ are orthonomal,that is ,$\vec u \vec u^H=I$,and $\vec v \vec v^H=I$.
So now $H=U\Sigma V^H$,and $y=Hs+n=(U\Sigma V^H)s+n$
and $U^H y=\Sigma V^Hs+n$,see the $\Sigma V^Hs$,isn't it like " channel gain $\times$ beamforming $\times$ signal $s$ "?
Now you can also find that according to the rule of multiplication of matrix,the first element of $\Sigma$,which is the $\sigma_1$,will multiple every element of $v_1^H$,that is why we say the $v^H_1$ is the best precoding ,because its elements will multiply the biggest singular value ,$\sigma_1$, every time.
