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Inverse function theorem over matrices

  1. Jun 6, 2014 #1
    Hi there!
    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] ?

    Last edited: Jun 6, 2014
  2. jcsd
  3. Jun 7, 2014 #2
    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 [itex]Y[/itex] is near the identity, then there exists only one [itex]X[/itex] such that [itex]X^2 = Y[/itex]" ? If so this is false.
  4. Jun 7, 2014 #3
    Thanks for your response MisterX.
    What I mean with "a new dot product" is defining a product between vectors that represents the matrix multiplication.

    Uhmm, yes that's it, it's an extra exercice they give me in my calculus II course. Why you say it's false?
  5. Jun 7, 2014 #4
    Then it belongs in the homework forums. Please post there.
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