## 2018年2月20日火曜日

### 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(固有多項式、求め方)

1. $\begin{array}{}\mathrm{det}\left(\begin{array}{cc}x-2& -1\\ -2& x-3\end{array}\right)\end{array}={x}^{2}-5x+4\\ =\left(x-1\right)\left(x-4\right)\\ =0$

よって、固有値は、

$x=1,4$

固有値1について。

$\left(\begin{array}{cc}2& 1\\ 2& 3\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=1\left(\begin{array}{c}x\\ y\end{array}\right)$
$\begin{array}{}2x+y=x\\ 2x+3y=y\\ x+y=0\\ 2x+2y=0\end{array}$

よって、固有ベクトルは、

$\begin{array}{}c\ne 0\\ c\left(\begin{array}{c}1\\ -1\end{array}\right)\end{array}$

固有値4について。

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

よって固有ベクトルは、

$\begin{array}{}c\ne 0\\ c\left(\begin{array}{c}1\\ 2\end{array}\right)\end{array}$

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, solve(eq)]:
pprint(t)
print()


$./sample25.py (x - 3)⋅(x - 2) - 2 [1, 4]$


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">

<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="sample25.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) => 2 * x + y,
fy = (x, y) => 2 * x + 3 * y,
g = (x) => -x,
h = (x) => 2 * x,
fns1 = [[g, 'red'],
[h, '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 = [];

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();