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Let $f\left(A\right)=e^{A^{2}}$ where $A$ is an $n\times n$ matrix. Show that $f$ is differentiable and compute its derivative.

I know this is kind of a basic question, but I am not sure how to solve it. By definition, we want to show that there exists a matrix $B$ such that $\lim_{H\to0}\frac{\left|f\left(A+H\right)-f\left(A\right)-BH\right|}{\left|H\right|}=0.$ Is this right? But since we're working with matrices I am not sure what to do. Any guidence would be appreciated.

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    The linear part in $H$ is rather complicated, but you can write it down as an infinite sum of terms like $\sum_{n\geq 0}\frac{1}{n!}\sum_{i=0,\dots,n} A^iHA^{n-i}$ if I'm not mistaken. You only need to show that indeed converges2012-05-09

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$Df(A):H\mapsto\sum_{n=1}^{+\infty}\frac1{n!}\sum_{i=1}^{2n}A^{i-1}HA^{2n-i}$