数学 - Python - 線形代数学 - 多項式と行列 - 特性多項式 - 複素固有値と固有ベクトル、因数分解

ラング線形代数学(下)
原著
楽天ブックス Yahoo!

学習環境

• Surface、Surface ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
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• 参考書籍
  • Pythonからはじめる数学入門 (楽天ブックス Yahoo!)
  • JavaScriptリファレンス 第6版 (楽天ブックス Yahoo!)
  • インタラクティブ・データビジュアライゼーション (楽天ブックス Yahoo!)

ラング線形代数学(下) (ちくま学現文庫)(S.ラング (著)、芹沢 正三 (翻訳)、筑摩書房)の9章(多項式と行列)、4(特性多項式)、練習問題3の解答を求めてみる。

3. 
   
   1. 
      

      特性多項式。

      $\begin{array}{l}f\left(t\right)\\ =\det \left(tI-A\right)\\ =\det \left[\begin{array}{cc}t-1& -i\\ -i& t+2\end{array}\right]\\ =t^2+t-2+1\\ =t^2+t-1\end{array}$

      固有値。

      $\begin{array}{l}{t}^2+t-1=0\\ t=\frac{-1\pm \; <!--\; plus-minus\; sign\; -->\sqrt{1+4}}{2}\\ =\frac{-1\pm \; <!--\; plus-minus\; sign\; -->\sqrt{5}}{2}\end{array}$

      それぞれの固有値をもつ固有ベクトル。

      $\begin{array}{l}\{\begin{array}{l}\left(t-1\right)x-iy=0\\ -ix+\left(t+2\right)y=0\end{array}\\ \frac{-3\pm \; <!--\; plus-minus\; sign\; -->\sqrt{5}}{2}x-iy=0\\ \left(2,\left(3\mp \; <!--\; latex:\; \backslash mp\; -->\sqrt{5}\right)i\right)\end{array}$
   2. 
      
      $\begin{array}{l}f\left(t\right)\\ =\det \left[\begin{array}{cc}t-1& -i\\ i& t-1\end{array}\right]\\ =t^2-2t+1-1\\ =t\left(t-2\right)\\ t=0,2\\ \{\begin{array}{l}\left(t-1\right)x-iy=0\\ ix+\left(t-1\right)y=0\end{array}\\ t=0\\ \{\begin{array}{l}-x-iy=0\\ ix-y=0\end{array}\\ \left(1,i\right)\\ t=2\\ \{\begin{array}{l}x-iy=0\\ ix+y=0\end{array}\\ \left(1,-i\right)\end{array}$
   3. 
      
      $\begin{array}{l}f\left(t\right)=\left(t-1\right)\left(t-2\right)\\ t=1,2\\ \{\begin{array}{l}\left(t-1\right)x-2iy=0\\ \left(t-2\right)y=0\end{array}\\ t=1\\ \left(1,0\right)\\ t=2\\ x-2iy=0\\ \left(2i,1\right)\end{array}$
   4. 
      
      $\begin{array}{l}f\left(t\right)=\left(t-2\right)\left(t-3\right)-20\\ =t^2-5t-14\\ =\left(t-7\right)\left(t+2\right)\\ t=-2,7\\ \{\begin{array}{l}\left(t-2\right)x-4y=0\\ -5x+\left(t-3\right)y=0\end{array}\\ t=-2\\ \{\begin{array}{l}-4x-4y=0\\ -5x-5y=0\end{array}\\ \left(1,1\right)\\ t=7\\ \{\begin{array}{l}5x-4y=0\\ -5x+4y=0\end{array}\\ \left(4,5\right)\end{array}$
   5. 
      
      $\begin{array}{l}f\left(t\right)=\left(t-1\right)\left(t+2\right)-4\\ =t^2+t-6\\ =\left(t+3\right)\left(t-2\right)\\ t=-3,2\\ \{\begin{array}{l}\left(t-1\right)x-2y=0\\ -2x+\left(t+2\right)y=0\end{array}\\ t=-3\\ \{\begin{array}{l}-4x-2y=0\\ -2x-y=0\end{array}\\ \left(1,-2\right)\\ t=2\\ \{\begin{array}{l}x-2y=0\\ -2x+4y=0\end{array}\\ \left(2,1\right)\end{array}$
   6. 
      
      $\begin{array}{l}f\left(t\right)={\left(t-3\right)}^2+4\\ =t^2-6t+13\\ t=3\pm \; <!--\; plus-minus\; sign\; -->\sqrt{9-13}\\ =3\pm \; <!--\; plus-minus\; sign\; -->2i\\ \{\begin{array}{l}\left(t-3\right)x-2y=0\\ 2x+\left(t-3\right)y=0\end{array}\\ t=3+2i\\ \{\begin{array}{l}2ix-2y=0\\ 2x+2iy=0\end{array}\\ \left(1,i\right)\\ t=3-2i\\ \{\begin{array}{l}-2ix-2y=0\\ 2x-2iy=0\end{array}\\ \left(1,-i\right)\end{array}$
   7. 
      
      $\begin{array}{l}f\left(t\right)\\ =\left(t+1\right)\left(t-2\right)\left(t+6\right)+12+24\\ +\left(t+1\right)12-4\left(t+6\right)+6\left(t-2\right)\\ =\left(t^2-t-2\right)\left(t+6\right)+36+12t+12-4t-24+6t-12\\ =t^3+5t^2-8t-12+12+14t\\ =t^3+5t^2+6t\\ =t\left(t^2+5t+6\right)\\ =t\left(t+2\right)\left(t+3\right)\\ t=-3,-2,0\\ \{\begin{array}{l}\left(t+1\right)x-2y-2z=0\\ -2x+\left(t-2\right)y-2z=0\\ 3x+6y+\left(t+6\right)z=0\end{array}\\ t=-3\\ \{\begin{array}{l}-2x-2y-2z=0\\ -2x-5y-2z=0\\ 3x+6y+3z=0\end{array}\\ z=-x-y\\ -3y=0\\ y=0\\ z=-x\\ \left(1,0,-1\right)\\ t=-2\\ \{\begin{array}{l}-x-2y-2z=0\\ -2x-4y-2z=0\\ 3x+6y+4z=0\end{array}\\ x=-2y-2z\\ 2z=0\\ z=0\\ x=-2y\\ \left(-2,1,0\right)\\ t=0\\ \{\begin{array}{l}x-2y-2z=0\\ -2x-2y-2z=0\\ 3x+6y+6z=0\end{array}\\ x=y+z\\ y+z+2y+2z=0\\ y+z=0\\ \left(0,1,-1\right)\end{array}$
   8. 
      
      $\begin{array}{l}f\left(t\right)\\ =\left(t-3\right)\left(t-1\right)\left(t+1\right)-\left(t-3\right)2\\ =\left(t-3\right)\left(t^2-1-2\right)\\ =\left(t-3\right)\left(t^2-3\right)\\ t=\pm \; <!--\; plus-minus\; sign\; -->\sqrt{3},3\\ \{\begin{array}{l}\left(t-3\right)x-2y-z=0\\ \left(t-1\right)y-2z=0\\ -y+\left(t+1\right)z=0\end{array}\\ t=-\sqrt{3}\\ \{\begin{array}{l}\left(-\sqrt{3}-3\right)x-2y-z=0\\ \left(-\sqrt{3}-1\right)y-2z=0\\ -y+\left(-\sqrt{3}+1\right)z=0\end{array}\\ z=-\frac{\sqrt{3}+1}{2}y\\ \left(-\sqrt{3}-3\right)x-2y+\frac{\sqrt{3}+1}{2}y=0\\ 2\left(-\sqrt{3}-3\right)x-4y+\left(\sqrt{3}+1\right)y=0\\ 2\left(-\sqrt{3}-3\right)x+\left(\sqrt{3}-3\right)y=0\\ x=\frac{\sqrt{3}-3}{2\left(\sqrt{3}+3\right)}y\\ z=-\frac{\sqrt{3}+1}{2}·\; <!--\; middle\; dot\; -->2\left(\sqrt{3}+3\right)\\ =-\left(\sqrt{3}+1\right)\left(\sqrt{3}+3\right)\\ =-4\sqrt{3}-6\\ =-2\left(\sqrt{3}+1\right)\\ \left(\sqrt{3}-3,2\left(\sqrt{3}+3\right),-2\left(\sqrt{3}+1\right)\right)\\ t=\sqrt{3}\\ \{\begin{array}{l}\left(\sqrt{3}-3\right)x-2y-z=0\\ \left(\sqrt{3}-1\right)y-2z=0\\ -y+\left(\sqrt{3}+1\right)z=0\end{array}\\ y=\left(\sqrt{3}+1\right)z\\ \left(\sqrt{3}-3\right)x-2\left(\sqrt{3}+1\right)z-z=0\\ \left(\sqrt{3}-3\right)x+\left(-2\sqrt{3}-3\right)z=0\\ x=\frac{2\sqrt{3}+3}{\sqrt{3}-3}z\\ y=\left(\sqrt{3}+1\right)\left(\sqrt{3}-3\right)\\ =3-3-2\sqrt{3}\\ =-2\sqrt{3}\\ \left(2\sqrt{3}+3,-2\sqrt{3},\sqrt{3}-3\right)\\ t=3\\ \{\begin{array}{l}-2y-z=0\\ 2y-2z=0\\ -y+4z=0\end{array}\\ \left(1,0,0\right)\end{array}$

コード

Python 3

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

print('3.')

print('(g)')

t = symbols('t')
I3 = eye(3)
A = Matrix([[-1, 2, 2],
            [2, 2, 2],
            [-3, -6, -6]])

f = (t * I3 - A).det()

for o in [f, f.factor()]:
    pprint(o)
    print()

入出力結果(Bash、cmd.exe(コマンドプロンプト)、Terminal、Jupyter(IPython))

C:\Users\...>py sample3.py
3.
(g)
14⋅t + (t - 2)⋅(t + 1)⋅(t + 6) + 12

t⋅(t + 2)⋅(t + 3)


C:\Users\...>