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Symmetry of generators

+2 votes
71 views
asked Mar 23, 2016 in STA 3045F - Adv. Stochastic Processes by Emiel_Zyde (680 points)
recategorized Mar 1 by Rowan

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 simon_rigby (4,220 points)
edited Apr 10 by simon_rigby
 
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 asilmotala (2,360 points)

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

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