Interpreting eigenvectors for an oscillatory Markov process

Question

Consider a 3-state system with states A, B, C such that at each discrete time step, the system advances with 100% certainty to the next state, e.g.

A - B - C - A - B...

The transition matrix for this system is

[0, 1, 0]
[0, 0, 1]
[1, 0, 0]

My intuition for this system says there are two modes:
1. A stationary mode with equal probability mass in each state
2. A periodic mode where the distribution repeats every 3 steps

Mode 1 should correspond to an eigenvalue of 1, which is guaranteed to exist for a Markov transition matrix. The eigenvector should be $[1/3, 1/3, 1/3]^T$

Mode 2 should correspond to a complex eigenvalue. The real and complex parts of the eigenvector should span the plane which intersects [1, 0, 0], [0, 1, 0], and [0, 0, 1].

However, complex eigenvalues always come in conjugate pairs for real matrices! This would seem to imply a periodic mode which is rotating in the opposite direction, and that doesn't make sense to me for this system. How should I interpret this 3rd eigenvalue?

Answer 1

I think you will see much more clearly what is happening if you find the two complex eigenvectors and let your matrix act upon each of them.

You will notice that both are 'rotating' in the same direction. When a real vector is a linear sum of these two vectors it then also rotates in this direction as you would expect.