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

Login

+3 votes

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.

- All categories
- BUS 1003H - Introduction to Financial Risk (39)
- BUS 2016H - Financial Mathematics (46)
- BUS 3018F - Models (69)
- BUS 3024S - Contingencies (47)
- BUS 4028F - Financial Economics (20)
- BUS 4027W - Actuarial Risk Management (20)
- BUS 4029H - Research Project (5)
- Mphil (1)
- Calculus and Pure Mathematics (3)
- Statistics (14)

...