(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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 :)

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Continuous square root function on the space of nxn matrices

**Physics Forums | Science Articles, Homework Help, Discussion**