But the ^-1 here is meaning the inverse matrix?
The Geometric Series formula also holds for matrices, when the eigenvalues of M are |λᵢ|<1pic.twitter.com/HV1uj8SWoA
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that's right
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This generalizes to a fun proof that 1 + (nilpotent object) = (invertible object) in a generic commutative ring. Just suppose M is nilpotent and the left series will eventually terminate!
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It also tells you how to differentiate 1/(some function whose derivative you know), since nilpotency of order 2 (i.e., dx*dx = 0) tells us that 1 /(1 + dx) = 1 - dx
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In any (not necessarily commutative) topological ring, whenever M^n tends to zero.
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Don't you need some completeness assumptions?
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And more generally in Banach algebras for elements with norm < 1
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IIRC this is a key step in proving Kantorovich’s theorem (sufficient conditions for convergence of Newton’s method)
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Is this a solution for linear algebra?
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