Why can we know the best precoding direction after we do the SVD to the channel?

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?

• You don't really need any theorem – $\sigma_1$ is exactly the factor with which the power is scaled. Nov 3, 2019 at 17:19

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.