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:

<br /> \begin{equation}<br /> M=\begin{bmatrix}<br /> S_1 &amp;C\\<br /> C^T &amp;S_2\\<br /> \end{bmatrix}<br /> \end{equation}<br />

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?

<br /> \begin{equation}<br /> M=\begin{bmatrix}<br /> 1 &amp;1\\<br /> 1 &amp;1\\<br /> \end{bmatrix}<br /> \end{equation}<br />
=> rank(M)=1
<br /> \begin{equation}<br /> M=\begin{bmatrix}<br /> 1 &amp;.5\\<br /> .5 &amp;1\\<br /> \end{bmatrix}<br /> \end{equation}<br />
=> 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