How does SVD work? Trying to find information, and, no-one seems to know the answers.
I have a time-series, represented by $T = [0, 1, 1, 0, \ldots, n]$ the time series is then transformed into the Spectral results:
$$
S = \begin{bmatrix}
 1&0 \\ 
 0&1 \\ 
 1&1 \\ 
 0&2 \\ 
 1&1 
\end{bmatrix}
$$
I then compute the SVD which gives me three resulting matrices u, s, v 
I understand that s contains the magnitudes that relate to the columns in both u and v but, how does this work for finding the most significant data within a given series? 
 A: The SVD of your matrix $S$ shows you in which direction $S$ amplifies most. You can construct the SVD in this spirit to understand this more:
Consider the problem $\|Sx\| \underset{\|x\| = 1}{\to} \max $, which is equivalent to $\|Sx\|^2 = x^T S^TSx \underset{\|x\| = 1}{\to} \max $. Applying the method of lagrange multiplicators yields the lagrange function $L(x, \sigma) = x^T S^TSx - \sigma (x^Tx - 1)$ where $\sigma$ is a lagrange multiplier. Now you get
$\nabla L(x,\sigma) = 0 \iff S^TSx -\sigma x = 0 \, \wedge \, x^Tx = 1 \iff S^T Sx = \sigma x \wedge x^T x = 1$.
You can see that the eigenvector of $S^TS$ corresponding to the lagrange multiplicator $\sigma$ gives the direction in which $S$ is amplifying most. 
You can now determine the $argmax$ by $x^TS^TSx = x^T\sigma x = \sigma$ and hence $\underset{\|x\| = 1}{argmax \|Sx\|} = \sqrt{\sigma} \,$  (the first singular value meaning the first diagonal entry of your matrix $s$). 
In the next Step you want to know in which direction $S$ amplifies "second most", but one has obviously to specify the term "second most". So let's name our already found direction $x_1$ and say we look in the orthogonal complement of $\{x_1\}$ for the direction in which $S$ amplifies most, and this is done in the same fashion like before. Only the lagrange function changes since we have now the additional constraint $\langle x,x_1\rangle = 0 \iff x^T x_1 = 0$ and by that we get the lagrange function $L(x, \sigma,\lambda) = x^T S^TSx - \sigma (x^Tx - 1) - \lambda x^T x_1$. This procedure is now iterated until we have $\min \{m,n \}$ singular values, where $S$ is a $m\times n$ matrix.
