# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 84: Some Properties of Transition Matrices (Probability)

January 4, 2017

Posted 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.

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