dc: without digging into the weeds, the norm condition tells you that the series makes sense and can be manipulated like a series. the rest is a series calculation. it's (I - H) (I + H + H^2 + ...) = I + H + H^2 + ... - (H + H^2 + H^3 + ...) = I
So the norm condition is only present for me to establish that the series is valid? i.e: I have to do that first in the way that I did then I can proceed with what you just suggested?
meaning I had to establish the $\frac{\|H\|}{1 - \|H\|}$ first?
yes. the particular upper bound you get on the norm of the series isn't as important as the fact that the series converges (in particular, that upper bound isn't used later)
i don't know what the exercise you're doing actually is, but yes, i'd hope so (i'm assuming it's something along the lines of: show that if |H| < 1 then I - H is invertible, maybe with or without the hint of using a series to construct the inverse)
I never encountered a scenario where a series would be the inverse to such an expression, granted I haven't done much work with matrices in this environment either. THe closest has been Cayley Hamilton stuff.
yeah, well in many cases the series will "simplify," i.e. the fact that you there's also some series expression for it might not be the most interesting thing. e.g. if H is an operator on a finite dimensional space, there's some N with every power of H being a polynomial of degree at most N, so the partial sums of the series are a sequence of polynomial approximants of degree at most N to an inverse that is also a polynomial in H of degree at most N
and there are ways of getting at that polynomial that do not detour through the series
so i'd think of the result as a very useful example of a technique (roughly: an analogy between matrices/operators and numbers, including via the algebra of polynomials and applying power series expansions of functions like 1/(1-x) to things that aren't numbers) but it is not, like, the final word on inverses, or even a suggestion that this series is always the most helpful thing
another way of approaching this is just, what happens if you take a beginning complex analysis book and write as many proofs as possible so that they only use the norm | | (in particular the triangle inequality) and the relation |zw| <= |z||w| between the norm and the multiplication
any results that survive that process are a kind of 'complex analysis' for anything in an algebra over C that has a norm like that
