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:
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.