The rank of a block matrix as a function of the rank of its submatrices.

[GoodSpirit]

Hello everyone,
I would like to post this problem here in this forum.
Having the following block matrix:
\(\displaystyle
M=\begin{bmatrix}
S_1 &C\\
C^T &S_2\\
\end{bmatrix}
\)

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


Can you help me?

I sincerely thank you!

All the best

GoodSpirit

 

[girodd]

why do you think the rank will be independent of C?

 

[GoodSpirit]

Hi girodd,

Thank you for answering!

Well it won´t! C is already related to S1 and S2.

It is a good idea to make the rank of M dependent of C too!

Thank you again!

All the best

Ricardo Sousa

 

[DrPhil]

Hello everyone,
I would like to post this problem here in this forum.
Having the following block matrix:
\(\displaystyle
M=\begin{bmatrix}
S_1 &C\\
C^T &S_2\\
\end{bmatrix}
\)

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

I would say that the maximum the rank could be is the sum of the ranks of S_1 and S_2, and the minimum is the larger of S_1 or S_2. The question is whether the variables represented by S_1 are independent of those in S_2. This form would be much easier to deal with if C = 0.

I found a detailed paper with a very general derivation, which I did not attempt to read. Your case is much simplified by the constraints on the four blocks - I suppose one could go through this paper step-by-step, simplifying as you go:
http://tesla.pmf.ni.ac.rs/people/Dragana/3.pdf

 

[GoodSpirit]

Hi Dr Phil,

Many thanks for your answer!
That paper you gave lead to the answer.
It lead to this paper:

"Equalities and inequalities for inertias of Hermitian
matrices with applications"

I need to apply it to some restrictions but I think I'm on the right way.

I really thank you

All the best

GoodSpirit