The rank of a block matrix as a function of the rank of its submatrice

Hello everyone,
I would like to post this problem here in this forum.
Having the following block matrix:

$$M=\begin{bmatrix} S_1 &C\\ C^T &S_2\\ \end{bmatrix}$$

I would like to find the function $f$ that holds $$rank(M)=f( rank(S1), rank(S2))$$.
$$S_1$$ and $$S_2$$ are covariance matrices-> symmetric and positive semi-definite.
$$C$$ is the cross covariance that may be positive semi-definite.

Can you help me?

I sincerely thank you! :)

All the best

GoodSpirit
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 Mentor Are you sure that this function exists? $$M=\begin{bmatrix} 1 &1\\ 1 &1\\ \end{bmatrix}$$ => rank(M)=1 $$M=\begin{bmatrix} 1 &.5\\ .5 &1\\ \end{bmatrix}$$ => rank(M)=2
 Hi mfb, Thank you for answering! :) True! it depends on something more! M is also a covariance matrix so C is related to S1 and S2. It is a good idea to make the rank M dependent of the C rank. The rank of M may be dependent of the eigen values that are shared by S1 and S2 Thankk you again All the best GoodSpirit