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

[GoodSpirit]

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

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

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

 

[mfb]

Are you sure that this function exists?

$$\begin{equation} M=\begin{bmatrix} 1 &1\\ 1 &1\\ \end{bmatrix} \end{equation}$$
=> rank(M)=1
$$\begin{equation} M=\begin{bmatrix} 1 &.5\\ .5 &1\\ \end{bmatrix} \end{equation}$$
=> rank(M)=2

 

[GoodSpirit]

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