Separation of inverse of invertible addition of matrices

Question

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.

Answer 1

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.

Answer 2

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 $M=\sum_{i=1}^\infty (-HQH^TR^{-1})^k$.