# What significance can be attached to the eigenmodes of a transition matrix?

I am studying a continuous dynamical system and am categorising preferred areas of the system's phase space using a 3 state hidden Markov model.

As the resulting transition matrix is row stochastic, it possesses three real eigenvalues, one of which is guaranteed to be 1. Naturally there are also three left eigenvectors of the matrix corresponding to these eigenvalues.

In my case I have a transition matrix:\begin{bmatrix}0.958&0.008&0.034\\0.000&0.987&0.013\\0.040&0.000&0.960\end{bmatrix}

with eigenvalues: \begin{bmatrix}1.00 & 0&0\\0&0.982&0\\0&0&0.923\end{bmatrix}

and left eigenvectors: \begin{align}\begin{bmatrix}1\\1\\1\end{bmatrix}; \begin{bmatrix}0.671\\0.147\\-0.727\end{bmatrix};\begin{bmatrix}0.191\\-0.919\\0.345\end{bmatrix}\end{align}

In the literature I have seen the leading eigenvector referred to as the stationary distribution of the states, while the other eigenvectors represent variations from the stationary distribution as modes in probability space, with decay timescales linked to the eigenvalues.

I am having trouble understanding this concept for two reasons. Firstly, as the eigenmodes can have negative elements I'm sure how to interpret them; they don't seem to represent probabilities of occcupying states. Nor do their elements sum to 0 which you might expect if these represented oscillations of probability from one state to another.

Secondly, as a Markov system has no memory, and I am not using probabilistic state allocation, I am not sure how the concept of a distribution over states enters into the dynamics of the process.

Any help understanding this issue would be much appreciated.