Eigenvalues in Discrete Time Markov chains?

I know that the matrices are not full rank and therefore one eigenvalue will be zero. I would like to find out if the non zero eigenvalues tell me anything about the system. Perhaps a double stochastic system might be of more interest? Anyways, does anyone out there know anything on this topic or have a good place to find out more info. My book and internet searches have not yielded anything useful. Thanks!