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Show that there is an $\epsilon > 0$ such that if $A$ is a real $2\times 2$ matrix satisfying $|a_{ij}|<\epsilon$, then there is a real $2\times2$ matrix $X$ such that $X^2+X^T=A$. Is $X$ unique?

I'm incredibly stuck here. I want to use the inverse function theorem here, where $f(X) = X^2+X^T$, but I don't know how to take the Jacobian of this matrix. Is this even the right direction?

Any help appreciated!

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    Perhaps do the $1\times 1$ case to get warmed up? – 2017-02-07
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    Yes, it's the right direction. Just pick any basis of $M_2(\mathbb R)$ and view $f$ as a function in four variables. E.g. if you pick the canonical basis $\{E_{11},E_{21},E_{12},E_{22}\}$, then $f'(0)=\pmatrix{1\\ &0&1\\ &1&0\\ &&&1}$. – 2017-02-08

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