数学 - Python - JavaScript - 図形の変換の方法 - 線形写像・1次変換 – 1次変換による色々な図形の像 - 固有値と固有ベクトル(不動点と不動直線)

数学読本〈6〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

数学読本〈6〉線形写像・1次変換/数論へのプレリュード/集合論へのプレリュード/εとδ/落ち穂拾い など(松坂 和夫(著)、岩波書店)の第22章(図形の変換の方法 - 線形写像・1次変換)、22.3(1次変換による色々な図形の像)、固有値と固有ベクトル、問26-(1).を取り組んでみる。

26. 
    
    1. 
       

       固有多項式。

       $\begin{array}{}\det (\begin{array}{cc}x-2& -1\\ -2& x-3\end{array})\end{array}=x^2-5x+4\\ =\left(x-1\right)\left(x-4\right)$

       固有値。

       $x=1,4$

       よって、

       $f\left(v\right)=v$

       国有値1に対する固有ベクトル。

       $(\begin{array}{cc}2& 1\\ 2& 3\end{array}\left)(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=(\begin{array}{c}{x}_1\\ x_2\end{array})$
       $\begin{array}{}2x_1+x_2=x_1\\ 2x_1+3x_2=x_2\end{array}$
       $\begin{array}{}{x}_1+x_2=0\\ 2x_1+2x_2=0\end{array}$
       $\begin{array}{}c\ne \; <!--\; not\; equal\; to\; -->0\\ c(\begin{array}{c}1\\ -1\end{array})\end{array}$

       ゆえに不動点は、直線

       $y=-x$

       上の点。

       国有値4に対する固有ベクトルは、

       $(\begin{array}{cc}2& 1\\ 2& 3\end{array}\left)(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=4(\begin{array}{c}{x}_1\\ x_2\end{array})$
       $\begin{array}{}2x_1+x_2=4x_1\\ 2x_1+3x_2=4x_2\end{array}$
       $\begin{array}{}2x_1-x_2=0\\ 2x_1-x_2=0\end{array}$
       $\begin{array}{}c\ne \; <!--\; not\; equal\; to\; -->0\\ c(\begin{array}{c}1\\ 2\end{array})\end{array}$

       よって不動直線は、

       $\begin{array}{}y=-x\\ y=2x+n\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Matrix, solve

x = symbols('x')
A = Matrix([[2, 1],
            [2, 3]])
E = Matrix([[1, 0],
            [0, 1]])
eq = (x * E - A).det()

for t in [eq, eq.factor(), solve(eq)]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample26.py
(x - 3)⋅(x - 2) - 2

(x - 4)⋅(x - 1)

[1, 4]

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.01">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<label for="n0">n = </label>
<input id="n0" type="number" value="1">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample26.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    input_n0 = document.querySelector('#n0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let fx = (x, y) => 2 * x + y,
    fy = (x, y) => 2 * x + 3 * y,
    f1 = (x) => - x,
    fns1 = [[f1, 'red']];
    

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        n0 = parseFloat(input_n0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        g1 = (x) => 2 * x + n0,
        lines = [],
        fns = [],
        fns2 = [];

    fns1.push([g1, 'green']);

    for (let x0 = x1; x0 <= x2; x0 += dx) {
        let fy0 = f1(x0),
            gy0 = g1(x0);
            
        points.push([fx(x0, fy0), fy(x0, fy0), 'blue']);
        points.push([fx(x0, gy0), fy(x0, gy0), 'orange']);
    }
    
    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });
    
    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), color]);
        });
    
    fns2
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), color]);
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r',(d) => d[3] ||  r)
        .attr('fill', (d) => d[2] || 'green');

    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    
    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 = n =