> * Kelly, Frank P. (1979). Reversibility and Stochastic Networks. Wiley, Chichester. pp. 21–25. By the way, this book is online [here](http://www.statslab.cam.ac.uk/~frank/BOOKS/book/whole.pdf)
That's funny, Frank Kelly was teaching me probability theory in 1979. Not from this book, though, we were learning more basic stuff.
Exercise 4, page 10 is
> Show that a stationary Markov chain is reversible if and only if the matrix of transition probabilities can be written as the product of a symmetric and a diagonal matrix.
The matrix of transition probabilities is not the rate matrix (=infinitesimal stochastic operator), but they're closely related.
Exercise 6 shows the eigenvalues can be real without reversibility.