[R] Relationship between covariance and inverse covariance matrices

[Vivian Shih]

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