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The matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.
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Efficient computation of matrix exponential of trace zero matrix
From the trace condition and Cayley-Hamilton it follows that the powers of $A$ satisfy a recurrence of the form
$$A^{n+3}= cA^{n+1}+(\det A) A^n$$,
for some explicit constant $c$. You actually …