## 2018年3月7日水曜日

### 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(不動点がなく不動直線がある場合)

1. 固有多項式。

$\begin{array}{}\mathrm{det}\left(\begin{array}{cc}x-5& 6\\ -3& x+4\end{array}\right)\end{array}={x}^{2}-x-2\\ =\left(x-2\right)\left(x+1\right)$

固有値。

$x=2,-1$

不動点なし。

固有ベクトル。

$\left(\begin{array}{cc}5& -6\\ 3& -4\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)=2\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)$
$\begin{array}{}5{x}_{1}-6{x}_{2}=2{x}_{1}\\ 3{x}_{1}-4{x}_{2}=2{x}_{2}\end{array}$
$\begin{array}{}3{x}_{1}-6{x}_{2}=0\\ 3{x}_{1}-6{x}_{2}=0\end{array}$
$\begin{array}{}c\ne 0\\ c\left(\begin{array}{c}2\\ 1\end{array}\right)\end{array}$
$\left(\begin{array}{cc}5& -6\\ 3& -4\end{array}\right)\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)=-\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right)$
$\begin{array}{}5{x}_{1}-6{x}_{2}=-{x}_{1}\\ 3{x}_{1}-4{x}_{2}=-{x}_{2}\end{array}$
$\begin{array}{}6{x}_{1}-6{x}_{2}=0\\ 3{x}_{1}-3{x}_{2}=0\end{array}$
$\begin{array}{}c\ne 0\\ c\left(\begin{array}{c}1\\ 1\end{array}\right)\end{array}$

よって、不動直線は、

$\begin{array}{}y=\frac{1}{2}x\\ y=x\end{array}$

コード(Emacs)

Python 3

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

x = symbols('x')
A = Matrix([[5, -6],
[3, -4]])
E = Matrix([[1, 0],
[0, 1]])
eq = (x * E - A).det()

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


$./sample26.py (x - 5)⋅(x + 4) + 18 (x - 2)⋅(x + 1) [-1, 2]$


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">

<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,
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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
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) => 5 * x - 6 * y,
fy = (x, y) => 3 * x - 4 * y,
f = (x) => 1 / 2 * x,
g = (x) => x,
fns1 = [[f, 'red'],
[g, 'green']];

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);

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

let points = [],
lines = [],
fns = [],
fns2 = [];

for (let x0 = x1; x0 <= x2; x0 += dx) {
let fy0 = f(x0),
gy0 = g(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])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();