Block of a covariance matrix

  • Thread starter GoodSpirit
  • Start date
  • #1
18
0

Main Question or Discussion Point

Hello everybody,

I’d like to present this math problem that I’ve trying to solve…
This matter is important because the covariance matrix is widely use and this leads to new interpretations of the cross covariance matrices.
Considering the following covariance block matrix :
[tex]
\begin{equation}
M=\begin{bmatrix}
S1 &C \\
C^T &S2 \\
\end{bmatrix}
\end{equation}
[/tex]

The matrix S1 and S2 are symmetric and positive semi-definite.C is also positive semi-definite
What I am trying figure out is :
1- I would like to discover the relation between the eigenvector of M and the eigen vectors of S1 and S2.
2- Discover the relation between the eigenvector of the matricez S1,S2 and C.
I used the eigendecomposition but it lead to a very complicated expressions…
Could you help me suggesting another approach?

I really thank you!

All the best

GoodSpirit
 

Answers and Replies

  • #2
13,214
10,111
Only possibility which I can think of, is to write the basis according to the blocks, and solve
$$
M\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}S_1& C\\ C^T&S_2 \end{bmatrix}\cdot \begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}S_1x+Cy\\C^Tx+S_2y\end{bmatrix} =\begin{bmatrix}\lambda x\\ \lambda y\end{bmatrix}
$$
which means to look for inverses of your blocks.
 

Related Threads on Block of a covariance matrix

  • Last Post
Replies
7
Views
2K
Replies
3
Views
5K
  • Last Post
Replies
2
Views
13K
  • Last Post
Replies
2
Views
3K
Replies
1
Views
9K
  • Last Post
Replies
4
Views
8K
Replies
1
Views
2K
Replies
15
Views
4K
Replies
1
Views
754
Replies
2
Views
4K
Top