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:

[tex]
\begin{equation}
M=\begin{bmatrix}
S_1 &C\\
C^T &S_2\\
\end{bmatrix}
\end{equation}
[/tex]

I would like to find the function $f$ that holds [tex]rank(M)=f( rank(S1), rank(S2))[/tex].
[tex]S_1[/tex] and [tex]S_2[/tex] are covariance matrices-> symmetric and positive semi-definite.
[tex]C[/tex] 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?

[tex]
\begin{equation}
M=\begin{bmatrix}
1 &1\\
1 &1\\
\end{bmatrix}
\end{equation}
[/tex]
=> rank(M)=1
[tex]
\begin{equation}
M=\begin{bmatrix}
1 &.5\\
.5 &1\\
\end{bmatrix}
\end{equation}
[/tex]
=> 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