TITLE: Inverse of sum of matrices rearrangement
QUESTION [1 upvotes]: I see this conversion of an inverse in a script and do not understand how this works. Could someone please explain or give me a hint please?
Would be so much appreciated.
$R W^T(WRW^T + Q)^{-1}$
= $(W^T Q^{-1} W + R^{-1})^{-1}W^TQ^{-1}$

REPLY [0 votes]: We'll use the property $(AB)^{-1} = B^{-1}A^{-1}$ a bit here.
We have:
$$\begin{split}&RW^T(WRW^T + Q)^{-1} \\
&= (W^{-T}R^{-1})^{-1}(WRW^{T} + Q)^{-1} \\
&= ((WRW^T+Q)(W^{-T}R^{-1}))^{-1}\\
&= (W+QW^{-T}R^{-1})^{-1}\\
&= (QW^{-T}(W^{T}Q^{-1}W+R^{-1}))^{-1}\\
&= (W^{T}Q^{-1}W+R^{-1})^{-1}(QW^{-T})^{-1}\\
&= (W^{T}Q^{-1}W+R^{-1})^{-1}W^{T}Q^{-1}\\\end{split}$$
Which is what we wanted.
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