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Let the space of all matrices over $\mathbb R$ of size $ n^2 $ , with the natural metric of $\mathbb R^{n^2 }$.

Prove that there exist a neighborhood of the identity matrix, such that all the matrices in that ball have a square root, i.e $$ \exists \varepsilon > 0\,:\forall X \in B\left( {I,\varepsilon } \right)\,\,\exists \,Y:\,Y^2 = X . $$

Hint: Consider the function $ F(X) = X^2 $ and use the Inverse Function Theorem

Help with this problem !! )=

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    Let $n=1$. Then not all matrices (= numbers) in the neighborhood of $0$ have a square root.2011-11-18
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    Wouldn't the following be a counterexample? Take any matrix with no square roots, and scale it (by a sufficiently small $\delta > 0$) so that the resulting matrix falls inside the ball.2011-11-18
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    Sorry!! The center of the ball is not the zero matrix, it´s the identity I2011-11-18
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    Compute the derivative of $F$ at the identity and show that it is invertible... Since $F(I) = I$ the inverse function theorem then does the rest.2011-11-18
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    How can I don´t know how to calculate the derivate of a matrix D:2011-11-18

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