title: 循環賽

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循環賽

題目

下圖為五位選手的比賽結果 a->b表示 a 獲勝

矩陣

$$A=\left[\begin{array}{ccccc}0& 1& 0& 1& 0\\ 0& 0& 1& 1& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 1\\ 1& 1& 1& 0& 0\end{array}\right]$$

$s^{(i)}$

$s^{(i)}=A{s}^{(i-1)}$ $$\begin{gathered} s^{(1)}=\left[\begin{array}{ccccc}0& 1& 0& 1& 0\\ 0& 0& 1& 1& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 1\\ 1& 1& 1& 0& 0\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}2\\ 2\\ 1\\ 2\\ 3\end{array}\right] \\ {s}^{(2)}=\left[\begin{array}{ccccc}0& 1& 0& 1& 0\\ 0& 0& 1& 1& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 1\\ 1& 1& 1& 0& 0\end{array}\right]\left[\begin{array}{c}2\\ 2\\ 1\\ 2\\ 3\end{array}\right]=2\left[\begin{array}{c}0\\ 0\\ 1\\ 0\\ 1\end{array}\right]+2\left[\begin{array}{c}1\\ 0\\ 0\\ 0\\ 1\end{array}\right]+1\left[\begin{array}{c}0\\ 1\\ 0\\ 1\\ 1\end{array}\right]+2\left[\begin{array}{c}1\\ 1\\ 0\\ 0\\ 0\end{array}\right]+3\left[\begin{array}{c}0\\ 0\\ 0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}4\\ 3\\ 2\\ 4\\ 5\end{array}\right] \end{gathered}$$

循環

$$A=\left[\begin{array}{ccccc}0& 1& 1& 0& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 1& 1\\ 1& 0& 0& 0& 1\\ 1& 1& 0& 0& 0\end{array}\right]$$

循環

$$\begin{gathered} s^{(1)}=\left[\begin{array}{ccccc}0& 1& 1& 0& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 1& 1\\ 1& 0& 0& 0& 1\\ 1& 1& 0& 0& 0\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}2\\ 2\\ 2\\ 2\\ 2\end{array}\right] \\ {s}^{(2)}=\left[\begin{array}{ccccc}0& 1& 1& 0& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 1& 1\\ 1& 0& 0& 0& 1\\ 1& 1& 0& 0& 0\end{array}\right]\left[\begin{array}{c}2\\ 2\\ 2\\ 2\\ 2\end{array}\right]=\left[\begin{array}{c}4\\ 4\\ 4\\ 4\\ 4\end{array}\right] \end{gathered}$$ $$As=\lambda s$$

$s^{(i)},i=?$

$$A=\left[\begin{array}{cccccc}0& 1& 1& 0& 0& 0\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 1& 1\\ 1& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 1& 0\end{array}\right]$$ $$s^{(1)}=\left[\begin{array}{c}2\\ 2\\ 2\\ 2\\ 1\\ 3\end{array}\right],s^{(2)}=\left[\begin{array}{c}4\\ 4\\ 3\\ 4\\ 2\\ 5\end{array}\right],s^{(3)}=\left[\begin{array}{c}7\\ 7\\ 6\\ 7\\ 4\\ 10\end{array}\right],s^{(4)}=\left[\begin{array}{c}13\\ 13\\ 11\\ 14\\ 7\\ 18\end{array}\right],s^{(5)}=\left[\begin{array}{c}24\\ 25\\ 21\\ 25\\ 13\\ 33\end{array}\right],s^{(6)}=\left[\begin{array}{c}46\\ 46\\ 38\\ 46\\ 24\\ 62\end{array}\right],s^{(7)}=\left[\begin{array}{c}84\\ 84\\ 70\\ 86\\ 46\\ 116\end{array}\right],$$