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
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Best Answer
You want to solve the equation $(I+X)^2=I+Y$ for the $(n\times n)$-matrix $X$ when $Y$ is a given matrix near $0$. This amounts to
$$Z(X):=2X+X^2\ =\ Y\ ,\qquad (*)$$
an equation in the $n^2$ unknowns $X_{ik}$. When $Y=0$ then $X=0$ is a solution of this equation. According to the assumptions of the implicit function theorem we have to check whether the $(n^2\times n^2)$-matrix
$$J:=\Bigl({\partial(Z_{11},Z_{12},\ldots,Z_{nn})\over \partial(X_{11},X_{12},\ldots,X_{nn})}\Bigl)_{X=0}$$
is regular. As $Z_{ik}=2 X_{ik} +$ terms of degree $2$ in the $X_{ik}$, the matrix $J$ is $2$ times the $n^2\times n^2$ identity matrix; so it is certainly regular. It follows that the equation $(*)$ has a solution $X=F(Y)$ such that $F(0)=0$ and $F(\cdot)$ is a differentiable matrix valued function of (the matrix elements of) $Y$.
By the way, the square root $I+X$ of $I+Y$ for $\|Y\|<1$ is given by the binomial series
$$\sum_{k=0}^\infty\ {1/2 \choose k}\ Y^k\ .$$