I am trying to understand a proof of the Eckart-Young Theorem (source in title). Let me add the definition and proof they provided, afterward I'll say what I do not understand.
Proof:
Some points I do not understand:
- Why does 4.98 holds?
- Couldn't the same/similar argument of "then there exists an at least $(n − k)$-dimensional null space of $\mathbf{B}$" also be made for $\hat{\mathbf{A}}(k)$ "there exists an exactly $(n − k)$-dimensional null space", since the rank is $k$? -> maybe I get that this proves that there can not be a matrix of smaller rank that better approximates it, but I do not understand how it proves that this $\hat{\mathbf{A}}(k)$ is the best matrix of rank $k$ that approximates it (probably because I do not understand why 4.98 holds?)