11.4.1, page 451 Define P and y by

\[ P = \left( \begin{array}{ccc} 0.5 & 0.5 \\ 0.25 & 0.75 \end{array} \right)\ , y = \left(\begin{array}{ccc} 1 \\ 0 \end{array} \right)\]

Compute \(Py\), \(P^2y\), and \(P^4y\) and show that the results are approaching a constant vector. What is this vector?

# compute Py
P <- matrix(c(0.5, 0.25, 0.5, 0.75), nrow = 2)
P
##      [,1] [,2]
## [1,] 0.50 0.50
## [2,] 0.25 0.75
y <- matrix(c(1,0))
y
##      [,1]
## [1,]    1
## [2,]    0
Py <- P %*% y
Py
##      [,1]
## [1,] 0.50
## [2,] 0.25
# compute P2y
P2y <- (P %*% P) %*% y
P2y
##        [,1]
## [1,] 0.3750
## [2,] 0.3125
# compute P4y
P4y <- (P %*% P %*% P %*% P) %*% y
P4y
##           [,1]
## [1,] 0.3359375
## [2,] 0.3320312


Just looking at the way the vectors are trending, it appears to be approaching \(\frac{1}{3}, \frac{1}{3}\), but I am not sure how to demonstrate this via a proof. I did go ahead and compute \(P^6y\) and \(P^8y\) to show that the vector continues to approach \(\frac{1}{3}, \frac{1}{3}\):

P6y <- (P %*% P %*% P %*% P %*% P %*% P) %*% y
P6y
##           [,1]
## [1,] 0.3334961
## [2,] 0.3332520
P8y <- (P %*% P %*% P %*% P %*% P %*% P %*% P %*% P) %*% y
P8y
##           [,1]
## [1,] 0.3333435
## [2,] 0.3333282