I have a matrix, $Z$, which happens to be diagonal. I am projecting it onto a lower-dimensional subspace, using a set of vectors, $\{y_i\}$. I write this as,
$$\xi_{ij} = \langle y_i| Z |y_j\rangle$$
I know the spectrum of Z. I know that it runs over the values: $N-1, N-2, ... ,1-N$. Here $N$ is related to the size of $Z$, but it is not the size itself.
I want to understand what can be said about the spectrum of the matrix $\xi_{ij}$, if anything. Does the fact that $Z$ is diagonal enable us to say more about the spectrum of $\xi_{ij}$ than we would be able to otherwise?
I feel like I'm touching on some fundamental points about projections of matrices, but I had a hard time finding a nice reference. If you can recommend a specific text chapter, this would also be appreciated.