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

block matriceslinear algebramatrices

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

\begin{equation}
M=\begin{bmatrix}
S_1 &C\\
C^T &S_2\\
\end{bmatrix}
\end{equation}

I would like to find the function $f$ that holds $\operatorname{rank}(M)=f( S_1, S_2,C)$
$S_1$ and $S_2$ are covariance matrices $\implies$ 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

Best Answer

Edit: This is impossible. The rank of $M$ is not uniquely determined by the ranks of its subblocks. For example, consider $$ M=\left[\begin{array}{cc|cc} 2&1&0&0\\ 1&1&0&\varepsilon\\ \hline 0&0&2&1\\ 0&\varepsilon&1&1\\ \end{array}\right]. $$ where $0<\varepsilon\le 1/2$. Then $M,S_1,S_2,C$ are positive semidefinite, $\operatorname{rank}(S_1)=\operatorname{rank}(S_2)=2$ and $\operatorname{rank}(C)=1$, but $\operatorname{rank}(M)=4$ when $\varepsilon<1/2$ and $\operatorname{rank}(M)=3$ when $\varepsilon=1/2$.

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