Question
Consider the regular transition matrix \[ A=\frac{1}{15}\left[\begin{array}{lllll} 4 & 2 & 5 & 1 & 3 \\ 1 & 3 & 4 & 5 & 2 \\ 3 & 5 & 1 & 2 & 4 \\ 2 & 1 & 3 & 4 & 5 \\ 5 & 4 & 2 & 3 & 1 \end{array}\right] \] Note that the matrix $15 A$ contains each of the integers $1,2,3,4,$ and 5 once in every row and in every column. a. Using technology, compute a high power of $A$, such as $A^{20} .$ What do you observe? Make a co
Consider the regular transition matrix \[ A=\frac{1}{15}\left[\begin{array}{lllll} 4 & 2 & 5 & 1 & 3 \\ 1 & 3 & 4 & 5 & 2 \\ 3 & 5 & 1 & 2 & 4 \\ 2 & 1 & 3 & 4 & 5 \\ 5 & 4 & 2 & 3 & 1 \end{array}\right] \] Note that the matrix $15 A$ contains each of the integers $1,2,3,4,$ and 5 once in every row and in every column. a. Using technology, compute a high power of $A$, such as $A^{20} .$ What do you observe? Make a conjecture for $\lim _{t \rightarrow \infty} A^{t} .($ In part e, you will prove this conjecture.) b. Use technology to find the complex eigenvalues of A. Is matrix $A$ diagonalizable over $\mathbb{C} ?$ c. Find the equilibrium distribution $\vec{x}_{e q u}$ for $A,$ that is, the unique distribution vector in the eigenspace $E_{1}$ d. Without using Theorem 7.4 .1 (which was proven only for matrices that are diagonalizable over $\mathbb{R}$ ), show that $\lim _{t \rightarrow \infty}\left(A^{t} \vec{x}_{0}\right)=\vec{x}_{e q u}$ for any distribution vector $\vec{x}_{0} .$ Hint: Adapt the proof of Theorem 7.4 .1 to the complex case. e. Find $\lim _{t \rightarrow \infty} A^{t},$ proving your conjecture from part a.
Answers
(a) Show that the matrix
$\mathbf{A}=\left[ \begin{array}{ll}{1} & {-1} \\ {4} & {-3}\end{array}\right]$
has the repeated eigenvalue $r=-1$ and that all the eigenvectors are of the form $\mathbf{u}=s \operatorname{col}(1,2)$
(b) Use the result of part (a) to obtain a nontrivial solu-
$\quad$ tion $\mathbf{x}_{1}(t)$ to the system $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$
(c) To obtain a second linearly independent solution to $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$ try $\mathbf{x}_{2}(t)=t e^{-t} \mathbf{u}_{1}+e^{-t} \mathbf{u}_{2}$ [Hint: Substitute $\mathbf{x}_{2}$ into the system $\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}$ and derive the relations
$$ (\mathbf{A}+\mathbf{I}) \mathbf{u}_{1}=\mathbf{0}, \quad(\mathbf{A}+\mathbf{I}) \mathbf{u}_{2}=\mathbf{u}_{1} $$
since $\mathbf{u}_{1}$ must be an eigenvector set $\mathbf{u}_{1}=$ $\operatorname{col}(1,2)$ and solve for $\mathbf{u}_{2} . ]$
(d) What is $(\mathbf{A}+\mathbf{I})^{2} \mathbf{u}_{2} ?$ (In Section $9.8, \mathbf{u}_{2}$ will be identified as a generalized eigenvector.)
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