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# Consider the regular transition matrix $A=\frac{1}{15}\left[\begin{array}{lllll} 4 & 2 & 5 & 1 & 3 \\ 1 & 3 & 4 & 5 & 2 \\ 3 &... ## 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.

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