I am running into some trouble proving this statement:
There is a square-root function in a neighborhood of the identity matrix in .
Basically I need to show that there is a function from a neighborhood of into , with , such that for all .
I know that I can apply the inverse-function theorem with . But I am unsure where to begin.
Although I like Opalg's solution considering it gives an explicit formula for I think you're suggestion is easier. Namely, consider by one can check that so by the uniqueness of the derivative (in terms of it's linear approximation) we may conclude that and so in particular which is invertible so that there exists some neighborhood of such that is invertible and has a inverse . Can you show that it is now ?