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I'm currently reading a paper where they use the matrix inversion lemma in a proof and I don't understand how they are applying it. The paper is ref and the proof is in Appendix A (page 13) (see image 2). I don't really understand the steps, but I'm stuck at the first one which is:

$ L = RR^T(RR^T+\gamma I)^{-1}\\ \;\: = I - \gamma(RR^T+\gamma I)^{-1}$

They're supposedly using the MIL but I can't manage to reproduce it.

1 Answers 1

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It's just the distributive law.

$$I = (RR^T + \gamma I)(RR^T + \gamma I)^{-1} = RR^T (RR^T + \gamma I)^{-1} + \gamma (RR^T + \gamma I)^{-1} $$

Now subtract $\gamma (RR^T +\gamma I)^{-1}$ from both sides.