$$(I-A)^{-1}$$ trick


The geometric series equals if and only if .

By squinting at the notation, we can notice (not prove) a “well-known” matrix identity. (I keep forgetting it, so it’s not well-known to me).

If we let and make the following replacements:

(where is some matrix and is the identity matrix), we can get the following “identity”:

Now we interpret “1 over something” as its multiplicative inverse, so the right hand side becomes .

Therefore, by our “identity”,

which is actually true.

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