Separation of inverse of invertible addition of matrices Consider the following $n\times n$ matrix:
$A^T (HQH^T + R)^{-1} A$.
I would like to know if there exists a matrix $M$, such that:
$A^T M A +  A^T R^{-1} A$
If so, what is its definition?
Some background information:


*

*$A=H\Theta$ is $m\times n$ 

*$\Theta$ is $n\times n$ 

*$H$ is $m\times n$ and has full rank
$n$, as $m>n$. 

*$R$ is $m\times m$, diagonal and invertible 

*$Q$ is $n\times n$ and positive semidefinite.


These conditions arise, as this term represents the inverse of a covariance matrix, which is composed by different error sources.
 A: At the end of the day, your question boils down to finding $(B+I)^{-1}$. You can use Neumann series:
$$
(I+B)^{-1}=\sum_{i=0}^\infty (-B)^k,
$$
if the series is convergent. You need then to show that a norm of $B=(HQH^TR^{-1})$  is smaller than one and then $M=\sum_{i=1}^\infty (-HQH^TR^{-1})^k$.
A: Thanks for your comments, however I just found the desired answer on my own using this post: https://math.stackexchange.com/q/75389
and this paper: http://dspace.library.cornell.edu/bitstream/1813/32750/1/BU-647-M.version2.pdf
Using equation 23 or 24 (for the reduced order of the dimension of the inverse) of the paper, $A^T(HQH^T+R)^{-1}A$ can be shown to be: $A^T R^{-1} A + A^T M A$ (as desired), where:
$M=-R^{-1} H (I + QH^T R^{-1} H)^{-1} Q H^T R^{-1}$
If you are additionally interested in $[A^T(HQH^T+R)^{-1}A]^{-1} = [A^T R^{-1} A + A^T M A]^{-1}$, equation 23/24 can be applied a second time, if you define $P=A^T R^{-1} A$. This has no performance advantages, but allows for interesting comparisons.
No approximations or infinite sums are involved in these expressions and they conform to the given boundary conditions.
