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

• GoodSpirit
In summary, a user is seeking help in finding a function that holds the rank of a block matrix M to be dependent on the ranks of S1 and S2, which are covariance matrices. The user also mentions that C, the cross covariance, may also play a role in this function. They provide two examples of M, one with a rank of 1 and the other with a rank of 2, and discuss the potential influence of eigenvalues shared by S1 and S2 on the rank of M.
GoodSpirit
Hello everyone,
I would like to post this problem here in this forum.
Having the following block matrix:

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

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

Last edited:

Are you sure that this function exists?

$$M=\begin{bmatrix} 1 &1\\ 1 &1\\ \end{bmatrix}$$
=> rank(M)=1
$$M=\begin{bmatrix} 1 &.5\\ .5 &1\\ \end{bmatrix}$$
=> rank(M)=2

Hi mfb,

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

## 1. What is a block matrix?

A block matrix is a matrix that is composed of smaller matrices, called submatrices, arranged in a rectangular grid. It is often used to simplify calculations and represent complex systems.

## 2. How is the rank of a block matrix determined?

The rank of a block matrix is determined by the ranks of its submatrices. Specifically, the rank of a block matrix is equal to the maximum rank of its submatrices. In other words, the rank of a block matrix is limited by the lowest rank of any of its submatrices.

## 3. Can a block matrix have a higher rank than its submatrices?

No, the rank of a block matrix cannot be higher than the rank of its submatrices. This is because the submatrices are the building blocks of the block matrix and their ranks ultimately determine the overall rank of the block matrix.

## 4. How does the rank of a block matrix affect its invertibility?

The rank of a block matrix plays a crucial role in its invertibility. A block matrix is invertible if and only if all of its submatrices are invertible. This means that if any of the submatrices have a rank of less than the total number of rows or columns, the block matrix will not be invertible.

## 5. Can the rank of a block matrix change if its submatrices are altered?

Yes, the rank of a block matrix can change if its submatrices are altered. The rank of a block matrix is dependent on the ranks of its submatrices, so any changes to the submatrices can potentially affect the overall rank of the block matrix.

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