The classical proof of the existence of the SVD factorization by Trefethen and Bau reports
Set $\sigma_1 = \mid\mid A \mid\mid_2$. By a compactness argument, there must be a vector $v_1 \in \mathbb{C}^n$ with $\mid\mid v_1 \mid\mid = 1$ and $\mid\mid u_1 \mid\mid_2 = \sigma_1 $ where $u_1 = A v_1$.
where $A$ is a complex matrix of size $m \times n$.
Because it is presented in such a brisk fashion, I expect it to be something very elementary, but I cannot follow the reasoning at all. I guess that we are interested in the compactness of $\mathbb{C}^n$, but what are the implications of compactness which are relevant in this case?
Thanks!
Best Answer
It depends on how the induced matrix norm was defined. I don't have the book handy, but I expect some pages earlier the authors to have put $$\|A\|_2=\sup_{\|v\|_2=1}\|Av\|_2.$$ Note that a function's sup is not always achieved (think of $1-\exp x$ ($x$ real) and $1$). Compactness (for your purposes here) is a quick way of saying that the sup is achieved by a vector $v$, i.e., there is a specific vector $v_1$ which satisfies $$\|v_1\|_2=1\text{ and }\|Av_1\|=\|A\|_2.$$