I am looking for some clarifications in the uniqueness portion of the proof of Theorem 4.1 of Trefethen & Bau's Numerical Linear Algebra.
The definition of the SVD and the proof exerpt from the text book are below.
The question I have is this - why does the existence of $y$ such that $||y||=1$ and $||By||=\sigma_{1}$ imply that $y$ is a singular vector?
This does not seem to imply that $By = \sigma_{1}u_{1}$ (where $u_{1}$ is the left singular vector corresponding to $\sigma_{1}$ (or at least I can't see why the norm conditions imply that).
A clearer and more detailed explanation of this would be very appreciated.
