##### Activity4.5.3

Consider the matrices

\begin{equation*} A=\left[\begin{array}{rr} 0 \amp 1 \\ 1 \amp 0 \\ \end{array}\right],\qquad B=\left[\begin{array}{rr} 0.4 \amp 0.3 \\ 0.6 \amp 0.7 \\ \end{array}\right] \text{.} \end{equation*}Verify that both \(A\) and \(B\) are stochastic matrices.

Find the eigenvalues of \(A\) and then find a steady-state vector for \(A\text{.}\)

We will form the Markov chain beginning with the vector \(\xvec_0 = \twovec{1}{0}\) and defining \(\xvec_{k+1} = A\xvec_k\text{.}\) The Sage cell below constructs the first \(N\) terms of the Markov chain with the command

What do you notice about the Markov chain? Does it converge to the steady-state vector for \(A\text{?}\)`markov_chain(A, x0, N)`. Define the matrix \(A\) and vector \(x0\) and evaluate the cell to find the first 10 terms of the Markov chain.Now find the eigenvalues of \(B\) along with a steady-state vector for \(B\text{.}\)

As before, find the first 10 terms in the Markov chain beginning with \(\xvec_0 = \twovec{1}{0}\) and \(\xvec_{k+1} = B\xvec_k\text{.}\) What do you notice about the Markov chain? Does it converge to the steady-state vector for \(B\text{?}\)

What condition on the eigenvalues of a stochastic matrix will guarantee that a Markov chain will converge to a steady-state vector?