Chapter 5.2, Problem 27E

Let A = $\left[\begin{array}{ccc}.5& .2& .3\\ .3& .8& .3\\ .2& 0& .4\end{array}\right]$, v₁ = $\left[\begin{array}{c}.3\\ .6\\ .1\end{array}\right]$, v₂ = $\left[\begin{array}{c}1\\ -3\\ 2\end{array}\right]$, v₃ = $\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right]$, and w = $\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$.

a. Show that v₁,v₂, and v₃ are eigenvectors of A. [Note: A is the stochastic matrix studied in Example 3 of Section 4.9.]

b. Let x₀ be any vector in ℝ³ with nonnegative entries whose stun is 1. (In Section 4.9, x₀, was called a probability vector.) Explain why there are constants c₁, c₂, and c₃ such that x₀ = c₁v₁ + c₂v₂ + c₃v₃. Compute w^Tx₀, and deduce that c₁ = 1.

c. For k = 1, 2, ...,define x_k = A^kx₀, with x₀ as in part (b). Show that x_k → v₁ as k increases.