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I want to show directly from the definition of exp(X) that it is Frechet differentiable at every $X \in \mathbb{R^{n\times n}}$. The definition being exp$(X):= \sum \frac{X^k}{k!}$

I know continuous partial derivatives implies the function is frechet differentiable. However, this was proven in my notes in the the context of functions from an n-dimensional Euclidean space to an m-dimensional one and so I'm not confident in extending it to matrix functions (matrix-valued). Can I take partials as: $\frac{\partial(exp(X))}{X_{ij}}$ where the denominator is the i-jth entry of the matrix $X$. Then show that these are continuous and therefore exp$(X)$ is differentiable at $X$? If yes, then this 'object' is a matrix is it not? And so is continuous iff each of it's entries are continuous? And am I correct in thinking $\frac{\partial(X)}{X_{ij}}$ is the matrix with a $1$ in the i-jth position and $0$ elsewhere? I could then use this to show (by product rule?) the higher order terms in the sum are partially differentiable w.r.t $X_{ij}$ and hence so is their sum.

Is the Jacobian matrix of derivative of exp$(X)$ an $n^2 \times n^2$ matrix? Any comments are welcome, just want someone to point out mistakes or perhaps add something you might suspect I don't quite understand. Thank you.

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