###################################################################
#          MATRIX ALGEBRA IN R
###################################################################

#########################################
#several ways to construct a matrix object
#########################################


#create an empty matrix object
A=matrix(NA,nrow=2,ncol=2)
#Fill it with values
#acces each position
A[1,1]=2
A[1,2]=1
A[2,1]=3
A[2,2]=5

A[1,]=c(2,3)
A[2,]=c(3,5)
#or
#this is a 2x2 matrix
#so it has two columns and two rows
#we can use cbind or rbind R commands to
#bind two columns or two rows in a matrix respectivelly

#create a vector with the first row
R1=c(2,1)
#create a vector with the second row
R2=c(3,5)
#use rbind to join the two rows into a matrix
A.rbind=rbind(R1,R2)
#if we use cbind on R1 and R2 then we will need to transpose
#the matrix obtained as column bind of the two rows of A!!!
cbind(R1,R2)
A.cbind=t(cbind(R1,R2))

#Another way is just to simply turn a vector of all values into a matrix
#ncol is number of columns
A=matrix(c(2,1,3,5),ncol=2,byrow=TRUE)
#be careful here with byrow = TRUE or FALSE, if it is TRUE 
# we will obtain the matrix A
# if it is FALSE we will get transposed A 
transp.A=matrix(c(2,1,3,5),ncol=2)

##################################
#matrix multiplication
##################################

B=matrix(c(4,2,8,1),ncol=2,byrow=TRUE)

#multiply matrices A and B using %*%

D=A %*% B


##################################
#NOTE: Do not use * for matrix multiplication
#we can check what happens if we use * instead of %*%
#when we want to multiply matrices:
A*B
#it returns a matrix where elements on a matching positions
#in A and B are multiplied which is not a
#matrix multiplication
##################################


##################################
#matrix inversion
##################################

A.inv=solve(A)
#check if AxA.inv=I (identity matrix)
I=A %*% A.inv


##################################
#determinant
##################################
A
A.det=det(A)


########################################
#Markov Chain transition probability 
#matrix
########################################


P=matrix(c(0.3,0.7,0.4,0.6),ncol=2,byrow=T)
#prob that it will rain two days from today
P2=P%*%P
#prob that it will rain four days from today
P4=P2%*%P2
#prob that it will rain eight days from today
P8=P4%*%P4


#Not much difference between P4 and P8
#limiting probability that the process will be in state
#j after large number of transitions.
#Note that this value is not dependent on the initial value anymore
#pi_j=lim Pij^n, for n approaching inf