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The rank of a block matrix as a function of the rank of its submatrice 
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#1
Feb913, 07:58 AM

P: 19

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 semidefinite. [tex]C[/tex] is the cross covariance that may be positive semidefinite. Can you help me? I sincerely thank you! :) All the best GoodSpirit 


#2
Feb913, 11:52 AM

Mentor
P: 12,081

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 


#3
Feb1113, 09:23 AM

P: 19

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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