DISCLAIMER: The following is based on Tao’s amazing blog. I made turned his derivations into a game called “Guess the Identity”, which I presented to my former research group at DTU Compute.
Our starting point is the matrix determinant lemma:
Let’s derive an identity that you have all seen countless times. Write down your guesses as we go along! Keep revealing the spoilers below and update your answer.
Step 1
Perturbation by
:
Step 2
Linearise:
Step 3
Use the cyclic property of traces:
This holds for all !
Step 4
The result we’re looking for is:
The Sherman-Morrison-Woodbury identity!