PCA and diagonalization of the covariance matrix I am preparing for an upcoming exam and having looked at older exams I found one PCA related question I am having trouble understanding:

You have a dataset of $N$ two-dimensional points $\ y^t $. You want to
  perform PCA on the dataset. You have already estimated that the data
  is zero-mean and has the covariance matrix 
$$ S = \begin{bmatrix}10 & 6 \\ 6 & 10 \\\end{bmatrix}  $$
and you know that covariance matrix can be diagonalized as $\ C^TSC = D $, where 
$$ C = \begin{bmatrix} 1/\sqrt{2} & -1/\sqrt{2} \\1/\sqrt{2} & 1/\sqrt{2} \end{bmatrix}  $$ $$ D = \begin{bmatrix}16 & 0 \\0 & 4\end{bmatrix}  $$
  
  
*
  
*A) Explain how the matrices $C$ and $D$ are related to PCA? 
  
*B) In
  2-dimensional space, plot the PCA coordinates (direction of the largest
  and second largest variance of the data)  
  
*C) Define the principal
  components of the data by $z^t=C^Ty^t$. What is the covariance matrix
  of $z^t$?
  

I am having trouble understanding how to begin. Matrix $D$ seems to be a diagonalized $S$ (as the diagonal adds up to $20$) but how to perform any calculations on given data I have no clue. What computations should I perform in order to be able to plot the PCA coordinates?
 A: This comes a bit late, but for any other people looking for a simple intuitive (non-mathematical) idea about PCA: one way to took at it is as follows:
if you have a straight line in 2D, let's say the line y = x. In order to figure out what's happening, you need to keep track of these two directions. However, if you draw it, you can see that actually, there isn't much happening in the direction 45 degrees pointing 'northwest' to 'southeast', and all the change happens in the direction perpendicular to that. This means you actually only need to keep track of one direction: along the line.
This is done by rotating your axes, so that you don't measure along x-direction and y-direction, but along combinations of them, call them x' and y'.
That is exactly encoded in the matrix transformation above: you can see it as a matrix transformation, but equivalently as a rotation of the direction in which you measure. Now I will refer you to maths literature, but do try to think of it as directions in which you measure.
