Today I Learned

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

84: Some Properties of Transition Matrices (Probability)

The following are some miscellaneous related properties of transition matrices describing stochastic processes such as Markov chains.

A ‘transition matrix’ is a square matrix A with real entries in [0, 1] such that the sum of the entries of any given of A is equal to 1, the interpretation being that A_{i, j} is the probability of a stochastic process transitioning from state j to state i. A ‘regular’ matrix is a square matrix A such that A^n contains strictly positive entries for some n > 1.

For any transition matrix:

  1. Every transition matrix has 1 as an eigenvalue.

For regular transition matrices:

  1. The algebraic multiplicity of 1 as an eigenvalue of A is 1.
  2. The columns of \lim _{n \rightarrow \infty} A^n are identical, all being the unique A-invariant probability vector v such that the sum of the entries of v is 1 (because it’s a probability vector) and A v = v.
  3. With v defined as above, for any probability vector w\lim _{n \rightarrow \infty} A^n w = v. That is, the system or process described by A eventually converges to a distribution described by v, regardless of it’s initial state.
Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: