I have understood as far as marked with the red line. I am trying to understand the proof that there exists an SVD for all matrices. I don't understand why there is needed for an orthonormal extension of u and v. I understand that $U_1$ and $V_1$ are column vectors.
I understand part of equation 4.5 and what I understand is:
$A=U\Sigma V^*$
$U_1AV_1^*=\Sigma$
And thus: $U_1AV_1^*=\begin{bmatrix} \sigma_1 \\ 0 \end{bmatrix}$
Thus far I understand but why is b introduced and why is W* introduced?? What does the proof seek to do at this point?
Last part of the proof.
