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I'm back again with functions over matrices.

I have a function [itex]f : M_{n\times n} \to M_{n\times n} / f(X) = X^2[/itex].

Is valid the inverse function theorem for the [itex]Id[/itex] matrix? It talks about the Jacobian at the [itex]Id[/itex], but I have no idea how get a Jacobian of that function. Can I see that matrices as vectors and redefine the function as [itex]f : R^{n^2} \to R^{n^2} / f(x) = x^2[/itex] using a new dot product?

Also, how can I prove that if a matrix [itex]Y[/itex] is near to [itex]Id[/itex] then [itex]\exists ! X / X^2 = Y[/itex] ?

Thanks!

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# Inverse function theorem over matrices

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