(Image below) Image shows the principal component analysis of two points {(1,3), (3,1)}. u1 and u2 are drawn which are corresponding eigenvectors after PCA. The points follow a normal distribution N(0,0.5^2).
The text says that the eigenvector u1 is associated with both noisy and noise-free signal, while eigenvector u2 is associated with only noise component. I don't understand how u2 is associated with only noise?
The only piece of information that you need to distinguish between the points $(1,3)$ and $(3,1)$ is a component in the direction of $u_1$.
In particular: the centroid of our data is $\mu = (2,2)$ (or right around there, anyway). We can deduce whether a signal $s = (s_1,s_2)$ should be $(1,3)$ or $(3,1)$ by stating that a signal is $(3,1)$ if $(s - \mu)^Tu_1$ is negative, and $(3,1)$ if $(s - \mu)^Tu_1$ is positive.
To that effect, the $u_1$ component of the signal contains all of its informational content.
On the other hand, under ideal setting, $(s - \mu)^Tu_2$ should always be $0$. The only reason for any deviation from $0$ is noise. So, we say that the $u_2$ direction is only associated with noise.