library("markovchain")
library("igraph")
# Part A
stateNames = c("1","2","3","4","5","6","7","8","9")
MC = new("markovchain",transitionMatrix=matrix(c(0,1/3,0,1/3,1/3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1/2,1/2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,1/3,0,0,0,0,1/3,1/3,1/2,0,0,0,1/2,0,0,0,0,0,0,0,0,0,1/2,0,0,1/2,0,0,0,0,0,0,0,0,1),byrow=TRUE,nrow=9,dimnames=list(stateNames,stateNames)))

plot(MC)

# Part B
summary(MC)
#Unnamed Markov chain  Markov chain that is composed by: 
#Closed classes: 
#9 
#Transient classes: 
#1 4 7 
#2 
#3 5 6 8 
#The Markov chain is not irreducible 
#The absorbing states are: 9

#Part C
canonicForm(MC)
#Q
#  1         2         3         4         5   6 7         8
#1 0.0 0.3333333 0.0000000 0.3333333 0.3333333 0.0 0 0.0000000
#2 0.0 0.0000000 1.0000000 0.0000000 0.0000000 0.0 0 0.0000000
#3 0.0 0.0000000 0.0000000 0.0000000 0.5000000 0.5 0 0.0000000
#4 0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0 1 0.0000000
#5 0.0 0.0000000 0.0000000 0.0000000 0.0000000 1.0 0 0.0000000
#6 0.0 0.0000000 0.3333333 0.0000000 0.0000000 0.0 0 0.3333333
#7 0.5 0.0000000 0.0000000 0.0000000 0.5000000 0.0 0 0.0000000
#8 0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.5 0 0.0000000

#Part D
steadyStates(MC)
#     1 2 3 4 5 6 7 8 9
#[1,] 0 0 0 0 0 0 0 0 1
#these are normalized (left) eigenvectors to eigenvalue 1. Since the MC from Problem 7 has only one recurrent class, the stationary distribution #should be only zeroes and a 1 for the one recurrent state (state 9).

#Part E
simulation = rmarkovchain(n=100,object=MC, t0="1")
simulation
#  [1] "5" "6" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9"
# [19] "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9"
# [37] "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9"
# [55] "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9"
# [73] "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9" "9"
# [91] "9" "9" "9" "9" "9" "9" "9" "9" "9" "9"

# Trajectory ends up, as expected, in the recurrent class {9}