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Mathmos6

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## Homework Statement

Hi again all,

I've just managed to prove the existence (non-constructively) of a 'square root function' f on some open epsilon-ball about the identity matrix 'I' such that [itex][f(A)]^2=A\qquad \forall\, A \,\text{ s.t.}\, \|I-A\|<\epsilon[/itex] within M

_{n}, the space of n*n matrices (note that's f(A)^2, not f^2(A), so for example the identity function wouldn't work) - I used the inverse function theorem on A^2 to deduce its existence. However, I was wondering whether there exists a function f such that f

^{2}(A)=A [itex]\forall A \in M_n[/itex]? Or does there only exist such a function in a finite ball? What about for something like a cube root or a quintuple root function?

I was thinking perhaps some sort of compactness argument might work, but I couldn't reason anything in particular (and that's only if it isn't possible for the whole of [itex]M_n[/itex], otherwise compactness wouldn't work I don't suppose)

Many thanks for any help :)

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