markov-utils.jl

using LinearAlgebra

# A utility for computing the unique invariant probability
# vector for a finite, time homogeneous, irreducible, and aperiodic Markov chain
# from its transition probability matrix.
function invariantProb(P)
	PT = transpose(P)
	left_eigenvectors = nullspace(PT - I)
	normalized_left_eigenvector = left_eigenvectors[:, 1] / sum(left_eigenvectors[:, 1])
	return normalized_left_eigenvector
end

# A utility for computing expected step behavior
# of a transient communication class in a finite, time homogeneous
# Markov chain.
function subStochasticM(Q)
	return inv(I - Q) 	
end

P = [
  0 1/2 1/2 0;
  1/2 0 0 1/2;
  1/2 0 0 1/2;
  0 1/2 1/2 0;
]

p = invariantProb(P)
println(p)

Q = [
  1/3 2/3 0 0;
  1/4 1/2 1/4 0;
  1/4 1/4 1/4 1/4;
  1/4 0 1/4 1/4;
]

M = subStochasticM(Q)
println(sum(M[:, 1]))