[R] Relationship between covariance and inverse covariance matrices

[Bert Gunter]

What does this have to do with R? Is this homework?

I suggest you post to some sort of math list. Perhaps others will have
more specific suggestions where.

-- Bert

On Thu, Dec 8, 2011 at 2:42 PM, Vivian Shih <vivs at ucla.edu> wrote:
> 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
>
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-- 

Bert Gunter
Genentech Nonclinical Biostatistics

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