I'm having problems following the proof shown on page 5 here from the note of MIT OCW $18.335J$ introduction to numerical methods class.
I can't see how you get from eqn $1.28$ to $1.29$?
Can someone please break it down step by step?
I seem to be totally confused I don't know where the 0 vector comes from? I don't know where the $\omega$ comes from or where the B comes from?
The best I could come up with is that $u_1 = \frac{A v_1}{\sigma_1}$
Then $U^*AV = \frac{v_1^*A^*Av_1}{\sigma_1}$
This is clearly not the full solution for the upper left entry.
Thanks
Baz
There is a typo.
Rather than $u_1^*Av_1$, the left hand side should be $U_1^*AV_1$.
They define $S=U_1^*AV_1$ and equation $(1.29)$ claims that the first column has at most one non-zero entry at the very beginning followed by zero.
To see that the $(1,1)$-entry of $S$ is $\sigma_1$:
$$u_1^*Av_1=\frac{\|Av_1\|^2}{\sigma_1}=\frac{\sigma_1^2}{\sigma_1}=\sigma_1$$
Remark: if $\sigma_1=0$, the matrix is the zero matrix and existence of SVD for the zero matrix is trivial.
Now we show that the $(j,1)$-entry of $S$ is $0$ when $j>1$:
$$u_j^*Av_1=u_j^*(\sigma_1u_1)=\sigma_1(u_j^*u_1)=0$$
since $u_j$ and $u_1$ are orthogonal.
$w^*$ and $B$ are meant to denote the other part of the matrix of $S$.