Some of the things I've learned every day since Oct 10, 2016
84: Some Properties of Transition Matrices (Probability)
January 4, 2017Posted by on
The following are some miscellaneous related properties of transition matrices describing stochastic processes such as Markov chains.
A ‘transition matrix’ is a square matrix with real entries in such that the sum of the entries of any given of is equal to 1, the interpretation being that is the probability of a stochastic process transitioning from state to state . A ‘regular’ matrix is a square matrix such that contains strictly positive entries for some .
For any transition matrix:
- Every transition matrix has as an eigenvalue.
For regular transition matrices:
- The algebraic multiplicity of as an eigenvalue of is .
- The columns of are identical, all being the unique -invariant probability vector such that the sum of the entries of is (because it’s a probability vector) and .
- With defined as above, for any probability vector , . That is, the system or process described by eventually converges to a distribution described by , regardless of it’s initial state.