Solved: Consider a sequence of exchange matrices {E2, E3, E4, E5,...,En}, where E2 = 0 1

QUESTION:

Consider a sequence of exchange matrices \(\left\{E_{2}, E_{3}, E_{4}, E_{5}, \ldots, E_{n}\right\} \text {, }\), where

\(E_{2}=\left[\begin{array}{cc}
0 & \frac{1}{2} \\
1 & \frac{1}{2}
\end{array}\right], \quad E_{3}=\left[\begin{array}{ccc}
0 & \frac{1}{2} & \frac{1}{3} \\
1 & 0 & \frac{1}{3} \\
0 & \frac{1}{2} & \frac{1}{3}
\end{array}\right] \text {, }\)

\(E_{4}=\left[\begin{array}{cccc}
0 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\
1 & 0 & \frac{1}{3} & \frac{1}{4} \\
0 & \frac{1}{2} & 0 & \frac{1}{4} \\
0 & 0 & \frac{1}{3} & \frac{1}{4}
\end{array}\right], \quad E_{5}=\left[\begin{array}{ccccc}
0 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\
1 & 0 & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\
0 & \frac{1}{2} & 0 & \frac{1}{4} & \frac{1}{5} \\
0 & 0 & \frac{1}{3} & 0 & \frac{1}{5} \\
0 & 0 & 0 & \frac{1}{4} & \frac{1}{5}
\end{array}\right]\)

and so on. Use a computer to show that \(E_{2}^{2}>0_{2}, E_{3}^{3}>0_{3}, E_{4}^{4}>\mathbf{0}_{4}, E_{5}^{5}>\mathbf{0}_{5}\), and make the conjecture that although \(E_{n}^{n}>0_{n}\) is true, \(E_{n}^{k}>0_{n}\) is not true for k = 1, 2, 3,..., n ? 1. Next, use a computer to determine the vectors \(\mathbf{p}_{n}\) such that \(E_{n} \mathbf{p}_{n}=\mathbf{p}_{n}\) (for n = 2, 3, 4, 5, 6), and then see if you can discover a pattern that would allow you to compute \(\mathbf{p}_{n+1}\) easily from \(p_n\). Test your discovery by first constructing \(p_8\) from

\(\mathbf{p}_{7}=\left[\begin{array}{r}
2520 \\
3360 \\
1890 \\
672 \\
175 \\
36 \\
7
\end{array}\right]\)

and then checking to see whether \(E_{8} \mathbf{p}_{8}=\mathbf{p}_{8}\).