# Symmetry of generators

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recategorized Mar 1, 2017

How would one go about proving that a transition matrix generated by a symmetric generator Q is itself symmetric?

answered Mar 23, 2016 by (4,220 points)
edited Apr 10, 2017

$$P(t) = \exp(Qt)$$
It's easy to see that the sum of two symmetric matrices is symmetric. It's also true that a symmetric matrix raised to any integer power is symmetric. [In general, if $$A$$ and $$B$$ are symmetric $$n \times n$$ matrices, then $$AB$$ is symmetric if and only if $$AB = BA$$ - try prove it].
It follows from the definition of $$\exp(X)$$ that $$P(t)$$ is symmetric.