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Topic: Relationship between covariance and inverse covariance matrices
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vivs

Posts: 1
Registered: 12/8/11
Relationship between covariance and inverse covariance matrices
Posted: Dec 8, 2011 6:06 PM
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I've been trying to figure out a special set of covariance matrices that causes some symmetric zero elements in the inverse covariance matrix but am having trouble figuring out if that is possible.

Say, for example, matrix a is a 4x4 covariance matrix with equal variance and zero covariance elements, i.e.

[,1] [,2] [,3] [,4]
[1,] 4 0 0 0
[2,] 0 4 0 0
[3,] 0 0 4 0
[4,] 0 0 0 4

Now if we let a[1,2]=a[2,1]=3, then the inverse covariance matrix will have nonzero elements on the diagonals as well as for elements [1,2] and [2,1]. If we further let a[3,4]=a[4,3]=0.5 then the indices of the nonzero elements of the covariance matrix also matches those indices of the inverse.

The problem is, if any of the nonzero off-diagonal indices match, then the inverse covariance matrix will have non-matching nonzero elements. For example, if a[1,2]=a[2,1]=3 as before but now we'll let a[2,3]=a[3,2]=0.5, then a would be:

[,1] [,2] [,3] [,4]
[1,] 4 3.0 0.0 0
[2,] 3 4.0 0.5 0
[3,] 0 0.5 4.0 0
[4,] 0 0.0 0.0 4

The inverse covariance matrix is now:
[,1] [,2] [,3] [,4]
[1,] 0.58333333 -0.44444444 0.05555556 0.00
[2,] -0.44444444 0.59259259 -0.07407407 0.00
[3,] 0.05555556 -0.07407407 0.25925926 0.00
[4,] 0.00000000 0.00000000 0.00000000 0.25

If we let a[1,2] and a[2,3] be nonzero, then the inverse will create a nonzero [1,3]. Does that happen all the time? I've tried to write out the algebraic system of linear equations for a and a-inverse but couldn't come
up with anything.

Let me know if I'm not clear on anything. Basically I'd just like to see what type of covariance matrices will produce an inverse covariance matrix with some zero elements.



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