Relationship between covariance and inverse covariance matrices

Hi,

? 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.





Thanks,
Vivian