数学 - Python - 線形代数学 - 行列式 – 行列式の存在(3 × 3、一般化、帰納法、関数、行列式、導関数、微分)

ラング線形代数学(上)
原著

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)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
• 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

ラング線形代数学(上)(S.ラング (著)、芹沢 正三 (翻訳)、ちくま学芸文庫)の6章(行列式)、3(行列式の存在)、練習問題10.を取り組んでみる。

10. 
    

    3 x 3行列式の場合。

    $\begin{array}{l}\phi \left(t\right)=\left|\begin{array}{ccc}f\left(t\right)& g\left(t\right)& h\left(t\right)\\ f\text{'}\left(t\right)& g\text{'}\left(t\right)& h\text{'}\left(t\right)\\ f\text{'}\text{'}\left(t\right)& g\text{'}\text{'}\left(t\right)& h\text{'}\text{'}\left(t\right)\end{array}\right|\\ =f\left(t\right)\left(g\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)-g\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)\right)-\\ g\left(t\right)\left(f\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)\right)+\\ h\left(t\right)\left(f\text{'}\left(t\right)g\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\left(t\right)g\text{'}\left(t\right)\right)\\ \\ \phi \text{'}\left(t\right)=f\text{'}\left(t\right)\left(g\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)-g\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)\right)+\\ f\left(t\right)\left(g\text{'}\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)-g\text{'}\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)+g\text{'}\left(t\right)h\text{'}\text{'}\text{'}\left(t\right)-g\text{'}\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)\right)-\\ g\text{'}\left(t\right)\left(f\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)\right)-\\ g\left(t\right)\left(f\text{'}\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)+f\text{'}\left(t\right)h\text{'}\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)\right)+\\ h\text{'}\left(t\right)\left(f\text{'}\left(t\right)g\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\left(t\right)g\text{'}\left(t\right)\right)+\\ h\left(t\right)\left(f\text{'}\text{'}\left(t\right)g\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\text{'}\left(t\right)g\text{'}\left(t\right)+f\text{'}\left(t\right)g\text{'}\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\left(t\right)g\text{'}\text{'}\left(t\right)\right)\\ =f\left(t\right)\left(-g\text{'}\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)+g\text{'}\left(t\right)h\text{'}\text{'}\text{'}\left(t\right)\right)-\\ g\left(t\right)\left(-f\text{'}\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)+f\text{'}\left(t\right)h\text{'}\text{'}\text{'}\left(t\right)\right)+\\ h\left(t\right)\left(-f\text{'}\text{'}\text{'}\left(t\right)g\text{'}\left(t\right)+f\text{'}\left(t\right)g\text{'}\text{'}\text{'}\left(t\right)\right)\\ =\left|\begin{array}{ccc}f\left(t\right)& g\left(t\right)& h\left(t\right)\\ f\text{'}\left(t\right)& g\text{'}\left(t\right)& h\text{'}\left(t\right)\\ f\text{'}\text{'}\text{'}\left(t\right)& g\text{'}\text{'}\text{'}\left(t\right)& h\text{'}\text{'}\text{'}\left(t\right)\end{array}\right|\end{array}$

    一般化。

    $\begin{array}{l}\phi \left(t\right)=\left|\begin{array}{ccc}{f}_1& \cdots & f_n\\ f_1\text{'}& \cdots & f_n\text{'}\\ & & \\ f_1{}^{\left(n-1\right)}& \cdots & f_n{}^{\left(n-1\right)}\end{array}\right|\\ ={\displaystyle \sum _{i=1}^n{\left(-1\right)}^{i+j}{f}_i\left|\begin{array}{ccc}{f}_1\text{'}& \cdots & f_n\text{'}\\ & & \\ f_1{}^{\left(n-1\right)}& \cdots & f_n{}^{\left(n-1\right)}\end{array}\right|}\\ \\ \phi \text{'}\left(t\right)\\ ={\displaystyle \sum _{i=1}^n{\left(-1\right)}^{i+j}\left(f{\text{'}}_i\left|\begin{array}{ccc}{f}_1\text{'}& \cdots & f_n\text{'}\\ & & \\ f_1{}^{\left(n-1\right)}& \cdots & f_n{}^{\left(n-1\right)}\end{array}\right|+f_i\left|\begin{array}{ccc}{f}_1\text{'}& \cdots & f_n\text{'}\\ & & \\ f_1{}^n& \cdots & f_n{}^n\end{array}\right|\right)}\\ =\left|\begin{array}{ccc}{f}_1\text{'}& \cdots & f_n\text{'}\\ f_1\text{'}& \cdots & f_n\text{'}\\ & & \\ f_1{}^{\left(n-1\right)}& \cdots & f_n{}^{\left(n-1\right)}\end{array}\right|+\left|\begin{array}{ccc}{f}_1& \cdots & f_n\\ f_1\text{'}& \cdots & f_n\text{'}\\ & & \\ f_1{}^n& \cdots & f_n{}^n\end{array}\right|\\ =\left|\begin{array}{ccc}0& \cdots & 0\\ f_1\text{'}& \cdots & f_n\text{'}\\ & & \\ f_1{}^{\left(n-1\right)}& \cdots & f_n{}^{\left(n-1\right)}\end{array}\right|+\left|\begin{array}{ccc}{f}_1& \cdots & f_n\\ f_1\text{'}& \cdots & f_n\text{'}\\ & & \\ f_1{}^n& \cdots & f_n{}^n\end{array}\right|\\ =\left|\begin{array}{ccc}{f}_1& \cdots & f_n\\ f_1\text{'}& \cdots & f_n\text{'}\\ & & \\ f_1{}^n& \cdots & f_n{}^n\end{array}\right|\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Matrix, sin, cos, Function, Derivative

print('10.')
f = Function('f')
g = Function('g')
h = Function('h')
t = symbols('t')
m1 = Matrix([[f(t), g(t), h(t)],
             list(map(lambda f0: Derivative(f0(t), t, 1), [f, g, h])),
             list(map(lambda f0: Derivative(f0(t), t, 2), [f, g, h]))])

m2 = Matrix([[f(t), g(t), h(t)],
             list(map(lambda f0: Derivative(f0(t), t, 1), [f, g, h])),
             list(map(lambda f0: Derivative(f0(t), t, 3), [f, g, h]))])

pprint(m1)
pprint(m2)

d1 = m1.det()
pprint(d1)
d2 = m2.det()
pprint(d2)

d11 = Derivative(d1, t, 1)
pprint(d11)

a = d11.doit()
b = d2.doit()
pprint(a)
pprint(b)
print(a == b)

入出力結果(Terminal, IPython)

$ ./sample10.py
10.
⎡  f(t)       g(t)       h(t)   ⎤
⎢                               ⎥
⎢d          d          d        ⎥
⎢──(f(t))   ──(g(t))   ──(h(t)) ⎥
⎢dt         dt         dt       ⎥
⎢                               ⎥
⎢  2          2          2      ⎥
⎢ d          d          d       ⎥
⎢───(f(t))  ───(g(t))  ───(h(t))⎥
⎢  2          2          2      ⎥
⎣dt         dt         dt       ⎦
⎡  f(t)       g(t)       h(t)   ⎤
⎢                               ⎥
⎢d          d          d        ⎥
⎢──(f(t))   ──(g(t))   ──(h(t)) ⎥
⎢dt         dt         dt       ⎥
⎢                               ⎥
⎢  3          3          3      ⎥
⎢ d          d          d       ⎥
⎢───(f(t))  ───(g(t))  ───(h(t))⎥
⎢  3          3          3      ⎥
⎣dt         dt         dt       ⎦
                2                         2                         2         
     d         d               d         d               d         d          
f(t)⋅──(g(t))⋅───(h(t)) - f(t)⋅──(h(t))⋅───(g(t)) - g(t)⋅──(f(t))⋅───(h(t)) + 
     dt         2              dt         2              dt         2         
              dt                        dt                        dt          

                2                         2                         2      
     d         d               d         d               d         d       
g(t)⋅──(h(t))⋅───(f(t)) + h(t)⋅──(f(t))⋅───(g(t)) - h(t)⋅──(g(t))⋅───(f(t))
     dt         2              dt         2              dt         2      
              dt                        dt                        dt       
                3                         3                         3         
     d         d               d         d               d         d          
f(t)⋅──(g(t))⋅───(h(t)) - f(t)⋅──(h(t))⋅───(g(t)) - g(t)⋅──(f(t))⋅───(h(t)) + 
     dt         3              dt         3              dt         3         
              dt                        dt                        dt          

                3                         3                         3      
     d         d               d         d               d         d       
g(t)⋅──(h(t))⋅───(f(t)) + h(t)⋅──(f(t))⋅───(g(t)) - h(t)⋅──(g(t))⋅───(f(t))
     dt         3              dt         3              dt         3      
              dt                        dt                        dt       
  ⎛                2                         2                         2      
d ⎜     d         d               d         d               d         d       
──⎜f(t)⋅──(g(t))⋅───(h(t)) - f(t)⋅──(h(t))⋅───(g(t)) - g(t)⋅──(f(t))⋅───(h(t))
dt⎜     dt         2              dt         2              dt         2      
  ⎝              dt                        dt                        dt       

                   2                         2                         2      
        d         d               d         d               d         d       
 + g(t)⋅──(h(t))⋅───(f(t)) + h(t)⋅──(f(t))⋅───(g(t)) - h(t)⋅──(g(t))⋅───(f(t))
        dt         2              dt         2              dt         2      
                 dt                        dt                        dt       

⎞
⎟
⎟
⎟
⎠
                3                         3                         3         
     d         d               d         d               d         d          
f(t)⋅──(g(t))⋅───(h(t)) - f(t)⋅──(h(t))⋅───(g(t)) - g(t)⋅──(f(t))⋅───(h(t)) + 
     dt         3              dt         3              dt         3         
              dt                        dt                        dt          

                3                         3                         3      
     d         d               d         d               d         d       
g(t)⋅──(h(t))⋅───(f(t)) + h(t)⋅──(f(t))⋅───(g(t)) - h(t)⋅──(g(t))⋅───(f(t))
     dt         3              dt         3              dt         3      
              dt                        dt                        dt       
                3                         3                         3         
     d         d               d         d               d         d          
f(t)⋅──(g(t))⋅───(h(t)) - f(t)⋅──(h(t))⋅───(g(t)) - g(t)⋅──(f(t))⋅───(h(t)) + 
     dt         3              dt         3              dt         3         
              dt                        dt                        dt          

                3                         3                         3      
     d         d               d         d               d         d       
g(t)⋅──(h(t))⋅───(f(t)) + h(t)⋅──(f(t))⋅───(g(t)) - h(t)⋅──(g(t))⋅───(f(t))
     dt         3              dt         3              dt         3      
              dt                        dt                        dt       
True
$