数学 - Pytyhon - 線型代数学 - 行列による世界旅行 - ループ、長さ、逆回り(mod 5)

対話・おもしろ線形代数
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対話・おもしろ線形代数 (木村良夫(著)、現代数学社)の第3話(行列による世界旅行)、君も挑戦してみようの問1の解答を求めてみる。

1. 
   
   $\begin{array}{l}\left[\begin{array}{cc}0& 1\\ 2& 3\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 2\end{array}\right]\\ \to \left[\begin{array}{c}2\\ 1\end{array}\right]\\ \to \left[\begin{array}{c}1\\ 2\end{array}\right]\\ \to \left[\begin{array}{c}2\\ 3\end{array}\right]\\ \to \left[\begin{array}{c}3\\ 3\end{array}\right]\\ \to \left[\begin{array}{c}3\\ 0\end{array}\right]\\ \to \left[\begin{array}{c}0\\ 1\end{array}\right]\\ \to \left[\begin{array}{c}1\\ 3\end{array}\right]\\ \to \left[\begin{array}{c}3\\ 1\end{array}\right]\\ \to \left[\begin{array}{c}1\\ 4\end{array}\right]\end{array}$

   $\begin{array}{l}\to \left[\begin{array}{c}4\\ 4\end{array}\right]\\ \to \left[\begin{array}{c}4\\ 0\end{array}\right]\\ \to \left[\begin{array}{c}0\\ 3\end{array}\right]\\ \to \left[\begin{array}{c}3\\ 4\end{array}\right]\\ \to \left[\begin{array}{c}4\\ 3\end{array}\right]\end{array}$

   $\begin{array}{l}\to \left[\begin{array}{c}3\\ 2\end{array}\right]\\ \to \left[\begin{array}{c}2\\ 2\end{array}\right]\\ \to \left[\begin{array}{c}2\\ 0\end{array}\right]\\ \to \left[\begin{array}{c}0\\ 4\end{array}\right]\\ \to \left[\begin{array}{c}4\\ 2\end{array}\right]\end{array}$

   $\begin{array}{l}\to \left[\begin{array}{c}2\\ 4\end{array}\right]\\ \to \left[\begin{array}{c}4\\ 1\end{array}\right]\\ \to \left[\begin{array}{c}1\\ 1\end{array}\right]\\ \to \left[\begin{array}{c}1\\ 0\end{array}\right]\end{array}$

   よって長さ24のループを作る。

   逆回りのループをつくる行列、実際に確認。

   $\left[\begin{array}{cc}1& 3\\ 1& 0\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}1\\ 1\end{array}\right]\to \left[\begin{array}{c}4\\ 1\end{array}\right]\to \left[\begin{array}{c}2\\ 4\end{array}\right]\to \left[\begin{array}{c}4\\ 2\end{array}\right]\to \left[\begin{array}{c}0\\ 4\end{array}\right]$

   $\to \left[\begin{array}{c}2\\ 0\end{array}\right]\to \left[\begin{array}{c}2\\ 2\end{array}\right]\to \left[\begin{array}{c}3\\ 2\end{array}\right]\to \left[\begin{array}{c}4\\ 3\end{array}\right]\to \left[\begin{array}{c}3\\ 4\end{array}\right]$

   $\to \left[\begin{array}{c}0\\ 3\end{array}\right]\to \left[\begin{array}{c}4\\ 0\end{array}\right]\to \left[\begin{array}{c}4\\ 4\end{array}\right]\to \left[\begin{array}{c}1\\ 4\end{array}\right]\to \left[\begin{array}{c}3\\ 1\end{array}\right]$

   $\to \left[\begin{array}{c}1\\ 3\end{array}\right]\to \left[\begin{array}{c}0\\ 1\end{array}\right]\to \left[\begin{array}{c}3\\ 0\end{array}\right]\to \left[\begin{array}{c}3\\ 3\end{array}\right]\to \left[\begin{array}{c}2\\ 3\end{array}\right]$

   $\to \left[\begin{array}{c}1\\ 2\end{array}\right]\to \left[\begin{array}{c}2\\ 1\end{array}\right]\to \left[\begin{array}{c}0\\ 2\end{array}\right]\to \left[\begin{array}{c}1\\ 0\end{array}\right]$

コード

#!/usr/bin/env python3
from unittest import TestCase, main
from sympy import Matrix

vs1 = []
vs2 = []


class TestMod5Loop(TestCase):
    def test1(self):
        m = Matrix([[0, 1],
                    [2, 3]])
        u = Matrix([[1], [0]])
        vs1.append(u)
        v = (m * u) % 5
        vs1.append(v)
        i = 1
        while u != v:
            v = (m * v) % 5
            vs1.append(v)
            i += 1
        self.assertEqual(i, 24)

    def test2(self):
        m = Matrix([[1, 3],
                    [1, 0]])
        u = Matrix([[1], [0]])
        vs2.append(u)
        v = (m * u) % 5
        i = 1
        vs2.append(v)
        while u != v:
            v = (m * v) % 5
            vs2.append(v)
            i += 1
        self.assertEqual(i, 24)
        self.assertEqual(vs1, list(reversed(vs2)))


if __name__ == "__main__":
    main()

入出力結果(Zsh、PowerShell、Terminal、Jupyter(IPython))

% ./sample1.py -v
test1 (__main__.TestMod5Loop) ... ok
test2 (__main__.TestMod5Loop) ... ok

----------------------------------------------------------------------
Ran 2 tests in 0.041s

OK
%