# Symmetry of generators

+2 votes
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asked Mar 23, 2016
recategorized Mar 1

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

## 2 Answers

+3 votes
answered Mar 23, 2016 by (4,220 points)
edited Apr 10

Best answer
It's good enough to prove that the exponential of a symmetric matrix is symmetric, since:
$$P(t) = \exp(Qt)$$
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.

0 votes
answered Mar 24, 2016 by (2,570 points)

Could also use kolmogorovs forward and backward equations to show derivative matrix is symmetric, since qij=qji