I am trying to analyze the correlation between two sets of random variables. In particular, consider the disjoint sets of random variables $\mathcal{A}=\left\{ A_{0},\dots,A_{n_{a}-1}\right\} $ and $\mathcal{B}=\left\{ B_{0},\dots,B_{n_{b}-1}\right\} $, define their union set as $\mathcal{C}\equiv\mathcal{A}\cup\mathcal{B}$, then the covariance matrix of the random variables in $\mathcal{C}$ can be written as:
\begin{align*} {\rm Cov}\left(\mathcal{C},\mathcal{C}\right) & \equiv\begin{pmatrix}{\rm Cov}\left(\mathcal{A},\mathcal{A}\right) & {\rm Cov}\left(\mathcal{A},\mathcal{B}\right)\\ {\rm Cov}\left(\mathcal{A},\mathcal{B}\right) & {\rm Cov}\left(\mathcal{B},\mathcal{B}\right) \end{pmatrix},\\ {\rm Cov}\left(\mathcal{D},\mathcal{F}\right)_{ij} & \equiv{\rm Cov}\left(D_{i},F_{j}\right),D_{i}\in\mathcal{D}\text{ and }F_{j}\in\mathcal{F}. \end{align*}
here ${\rm Cov}\left(\mathcal{A},\mathcal{A}\right),{\rm Cov}\left(\mathcal{A},\mathcal{B}\right)$ and ${\rm Cov}\left(\mathcal{B},\mathcal{B}\right)$ are covariances matrices. Given that we know the matrix ${\rm Cov}\left(\mathcal{C},\mathcal{C}\right)$, is there any systematic criteria to characterize the correlation between the sets $\mathcal{A}$ and $\mathcal{B}$?
I tried to compare the eigenvalues of ${\rm Cov}\left(\mathcal{C},\mathcal{C}\right)$, and ${\rm Cov}\left(\mathcal{A},\mathcal{A}\right)$ and ${\rm Cov}\left(\mathcal{B},\mathcal{B}\right)$, and if there is no correlation between $\mathcal{A}$ and $\mathcal{B}$ then the spectrum ${\rm Cov}\left(\mathcal{C},\mathcal{C}\right)$ is equal to the union of the spectrum of ${\rm Cov}\left(\mathcal{A},\mathcal{A}\right)$ and ${\rm Cov}\left(\mathcal{B},\mathcal{B}\right)$. I want to know if there is a better way to quantify the correlation between the sets $\mathcal{A}$ and $\mathcal{B}$. Thanks.