Transition matrices are so called stochastic matrices, which means, that all entries are real values between zero and one, meaning, that each entry is a probability and a row sum equal one, meaning that the probability to jump to an arbitrary state is one
From the sum-statement follows easlily, that there is a right eigenvector with eigenvalue of one, which is constant
From the Perron-Frobenius Theorem follow immediately for stochastic matrices, thatImportant to know is, that the existance of the stationary distribution is not connected to the detailed balance property
Detailed balance is defined as
which is equivalent to a symmetry over the stationary distribution.
In some cases it is possible to postulate a generator, which taken to the exponent can construct the transition matrix for arbitrary timesteps.
This is only possibile (in a unique and intuitive way), if the transition matrix is positive definite
which is due to the logarithm of the eigenvalues, which are only uniquely defined, if the matrix is positive definite.
From the rate matrix we get back using