Chapter 5.2, Problem 25E

Let A = $\left[\begin{array}{cc}.6& .3\\ .4& .7\end{array}\right]$, v₁ = $\left[\begin{array}{c}3/7\\ 4/7\end{array}\right]$, x₀ = $\left[\begin{array}{c}.5\\ .5\end{array}\right]$. [Note: A is the stochastic matrix studied in Example 5 of Section 4.9.]

1. a. Find a basis for ℝ² consisting of v₁ and another eigenvector v₂ of A.
2. b. Verify that x₀ may be written in the form x₀ = v₁ + cv₂
3. c. For k = 1, 2, ...,define x_k = A^kx₀.Compute x₁ and x₂, and write a formula for x_k. Then show that x_k → v₁ as A increases.