Take a look at this link.
I am trying to understand the following source code meant for finding stationary distribution of a matrix:
# Stationary distribution of discrete-time Markov chain
# (uses eigenvectors)
stationary <- function(mat)
{
x = eigen(t(mat))$vectors[,1]
as.double(x/sum(x))
}
I tested the following source code myself:
> rm(list=ls())
>
> P <- matrix(c(0.66, 0.34,
+ 0.66, 0.34), nrow=2, ncol=2, byrow = TRUE)
>
> x <- eigen(t(P))
> x$values
[1] 1 0
$vectors
[,1] [,2]
[1,] 0.8889746 -0.7071068
[2,] 0.4579566 0.7071068
> y <- x$vectors[,1]
> y
[1] 0.8889746 0.4579566
>
looks like the command
y <- x$vectors[,1]
is selecting the 1st column of the matrix.
Why wasn't that simply written like the following?
# Stationary distribution of discrete-time Markov chain
# (uses eigenvectors)
stationary <- function(mat)
{
x = eigen(t(mat))
y = x[,1]
as.double(y/sum(y))
}
What was the reason for introduction of a dollar sign and vector keyword?
Let's test out your proposal:
> P <- matrix(c(0.66, 0.34, 0.66, 0.34), nrow=2, ncol=2, byrow = TRUE)
> x <- eigen(t(P))
> print(x)
eigen() decomposition
$values
[1] 1 0
$vectors
[,1] [,2]
[1,] 0.8889746 -0.7071068
[2,] 0.4579566 0.7071068
> y = x[,1]
This would produce the following error message:
Error in x[, 1] : incorrect number of dimensions
eigen returns a named list, with eigenvalues named values and eigenvectors named vectors. To access this component of the list. we use the dollar sign. Hence, that is why the code x$vectors which extract the matrix.