Largest positive eigenvalue of a matrix

Question

I am dealing with the Capacity of constrained noiseless communication channels. It has been said that the channel capacity of such a channel is $\log{\lambda}$, which $\lambda$ is the largest positive eigenvalue of the state transition matrix, however I wonder what if the largest positive value of such a matrix is complex. Can anyone tell the conditions which the matrix must satisfy till the largest positive eigenvalue of that matrix be real, not complex?

Answer 1

State-transition matrices necessarily have non-negative entries. By the Perron-Frobenius theorem, the eigenvalue with the greatest absolute value must be positive (and of course real) for any matrix with non-negative entries.