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

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SUMMARY

The discussion centers on determining the rank of a block matrix defined as M = [S1 C; C^T S2], where S1 and S2 are covariance matrices that are symmetric and positive semi-definite, and C is a cross covariance matrix. Participants agree that the rank of M is influenced by the ranks of S1, S2, and the relationship between them, particularly through shared eigenvalues. The conclusion emphasizes that the function f, which expresses rank(M) in terms of rank(S1) and rank(S2), must also account for the rank of C.

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Mathematicians, data scientists, and statisticians interested in matrix theory, particularly those working with covariance matrices and their applications in multivariate statistics.

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

[tex] \begin{equation}<br /> M=\begin{bmatrix}<br /> S_1 &C\\<br /> C^T &S_2\\<br /> \end{bmatrix}<br /> \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
 
Last edited:
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Are you sure that this function exists?

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

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