$\newcommand\mycolv[1]{\begin{bmatrix}#1\end{bmatrix}}$Hi all, I've been trying to learn all about Gaussian processes for a few days but I find myself stuck on some notation. The book that I have been mainly reading and other articles I've found split up a given multivariate Gaussian distribution into two random vectors $\textbf{x}$ and $\textbf{y}$ where $\textbf{y}$ contains the observed points. This makes sense to me until the redefinition of the gaussian comes up: $$\mycolv{\textbf{x} \\ \textbf{y}} \sim N(\mycolv{\mu_x \\ \mu_y}, \begin{bmatrix}\Sigma_{xx} & \Sigma_{xy} \\ \Sigma_{yx} & \Sigma_{yy}\end{bmatrix})$$
Unfortunately I am not clear on how to compute the values in this definition. Are the mean vectors just the mean of all of the distribution in each random vector i.e some scalar? Or are they a vector simply containing all of the means of each distribution contained in the corresponding vector?
As for the covariance matrix I can imagine having each $\Sigma$ be a covariance matrix, however, since this would cause the dimensions of the $\Sigma_{yx}$ and $\Sigma_{xy}$ matrices to be larger than the other matrices, making multiplication between those two matrices and the other two incompatible. This poses an issue since the product of $\Sigma_{yy}^{-1}$ and $\Sigma_{xy}$ is required when conditioning the multivariate.
Sorry if I made any errors. Any help would be appreciated.