Inverse function theorem over matrices

1. Jun 6, 2014

brunob

Hi there!
I'm back again with functions over matrices.
I have a function $f : M_{n\times n} \to M_{n\times n} / f(X) = X^2$.

Is valid the inverse function theorem for the $Id$ matrix? It talks about the Jacobian at the $Id$, but I have no idea how get a Jacobian of that function. Can I see that matrices as vectors and redefine the function as $f : R^{n^2} \to R^{n^2} / f(x) = x^2$ using a new dot product?
Also, how can I prove that if a matrix $Y$ is near to $Id$ then $\exists ! X / X^2 = Y$ ?

Thanks!

Last edited: Jun 6, 2014
2. Jun 7, 2014

MisterX

Yes, you can interpret the matrices as vectors. I'm not sure what you mean by "a new dot product". The square of a matrix would not be given by the dot product of the corresponding vector with itself; instead each component of resultant vector would be given by the formula for matrix multiplication.

Is this to be interpreted as "if $Y$ is near the identity, then there exists only one $X$ such that $X^2 = Y$" ? If so this is false.

3. Jun 7, 2014