TITLE: Proof of transpose property of matrix exponential
QUESTION [9 upvotes]: Using the fact that the matrix transpose distributes over infinite sums to show that $e^{(A^T)} = (e^A)^T$.  I feel like this is really trivial, but I don't know quite how to prove this.  How would I go about proving it?

REPLY [2 votes]: First, write out 
$$
e^{(A^T)} = I + \sum_1^\infty \frac{(A^T)^n}{n!} $$which does not require distributuivity of transpose, it is just exponentiation of an arbitrary matrix.
Then look at 
$$
e^A = I + \sum_1^\infty \frac{A^n}{n!} $$
And use distributivity of transpose and $(A^n)^T = (A^T)^N$to write
$$
(e^A)^T  = I + \sum_1^\infty \frac{(A^T)^n}{n!}
$$