- #1
brunob
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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!