Block of a covariance matrix

[GoodSpirit]

Hello everybody,

I’d like to present this math problem that I’ve trying to solve…
This matter is important because the covariance matrix is widely use and this leads to new interpretations of the cross covariance matrices.
Considering the following covariance block matrix :
$$\begin{equation} M=\begin{bmatrix} S1 &C \\ C^T &S2 \\ \end{bmatrix} \end{equation}$$

The matrix S1 and S2 are symmetric and positive semi-definite.C is also positive semi-definite
What I am trying figure out is :
1- I would like to discover the relation between the eigenvector of M and the eigen vectors of S1 and S2.
2- Discover the relation between the eigenvector of the matricez S1,S2 and C.
I used the eigendecomposition but it lead to a very complicated expressions…
Could you help me suggesting another approach?

I really thank you!

All the best

GoodSpirit

 

[fresh_42]

Only possibility which I can think of, is to write the basis according to the blocks, and solve
$$
M\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}S_1& C\\ C^T&S_2 \end{bmatrix}\cdot \begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}S_1x+Cy\\C^Tx+S_2y\end{bmatrix} =\begin{bmatrix}\lambda x\\ \lambda y\end{bmatrix}
$$
which means to look for inverses of your blocks.