- #1

- 15

- 0

## Homework Statement

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

**The questions**

Is valid the inverse function theorem for the identity matrix? It talks about the Jacobian at the identity, but I have no idea how get a Jacobian of that function. Can I see the 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 that represents the matrix multiplication?

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

Thanks!