数学 - Python - 線形代数学 - 行列式 - 行列式の存在 - 3次元、一般化、帰納法、全ての階数の導関数をもつ関数、行列式の導関数

ラング線形代数学(上)
原著
楽天ブックス Yahoo!

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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!)

ラング線形代数学(上) (ちくま学現文庫)(S.ラング (著)、芹沢 正三 (翻訳)、筑摩書房)の6章(行列式)、4(行列式の存在)、練習問題10の解答を求めてみる。

10. 
    
    $\frac{d}{\mathrm{dt}}\det \left[\begin{array}{ccc}f\left(t\right)& g\left(t\right)& h\left(t\right)\\ f\text{'}\left(t\right)& g\text{'}\left(t\right)& h\text{'}\left(t\right)\\ f\text{'}\text{'}\left(t\right)& g\text{'}\text{'}\left(t\right)& h\text{'}\text{'}\left(t\right)\end{array}\right]$

    $\begin{array}{l}=\frac{d}{\mathrm{dt}}f\left(t\right)\left(g\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)-g\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)\right)\\ -\frac{d}{\mathrm{dt}}g\left(t\right)\left(f\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)\right)\\ +\frac{d}{\mathrm{dt}}h\left(t\right)\left(f\text{'}\left(t\right)g\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\left(t\right)g\text{'}\left(t\right)\right)\end{array}$

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

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

    $=\det \left[\begin{array}{ccc}f\left(t\right)& f\left(t\right)& h\left(t\right)\\ f\text{'}\left(t\right)& g\text{'}\left(t\right)& h\text{'}\left(t\right)\\ f\text{'}\text{'}\text{'}\left(t\right)& g\text{'}\text{'}\text{'}\left(t\right)& h\text{'}\text{'}\text{'}\left(t\right)\end{array}\right]$

    一般化。

    $\frac{d}{\mathrm{dt}}\det \left[\begin{array}{ccc}{f}_1\left(t\right)& \dots & f_n\left(t\right)\\ f_1^{\left(1\right)}\left(t\right)& \dots & f_n^{\left(1\right)}\left(t\right)\\ ⋮& \text{none}& ⋮\\ f_1^{\left(n-1\right)}\left(t\right)& \dots & f_n^{\left(n-1\right)}\left(t\right)\end{array}\right]=\det \left[\begin{array}{ccc}{f}_1\left(t\right)& \dots & f_n\left(t\right)\\ f_1^{\left(1\right)}\left(t\right)& \dots & f_n^{\left(1\right)}\left(t\right)\\ ⋮& \text{none}& ⋮\\ f_1^{\left(n\right)}\left(t\right)& \dots & f_n^{\left(n\right)}\left(t\right)\end{array}\right]$

コード

#!/usr/bin/env python3
from unittest import TestCase, main
from sympy import Matrix, Function
from sympy.abc import t

print('10.')


class TestMatrixDetDerivative(TestCase):
    def test(self):
        n = 3
        f = Function('f')(t)
        g = Function('g')(t)
        h = Function('h')(t)
        a = Matrix([[f.diff(t, i), g.diff(t, i), h.diff(t, i)]
                    for i in range(n)]).det().diff(t, 1)
        b = Matrix(
            [[f.diff(t, i), g.diff(t, i), h.diff(t, i)]
             for i in range(n - 1)] +
            [[f.diff(t, n), g.diff(t, n), h.diff(t, n)]]).det()
        self.assertEqual(a.simplify(), b.simplify())


if __name__ == "__main__":
    main()

入出力結果(Zsh、PowerShell、Terminal、Jupyter(IPython))

% ./sample10.py -v
10.
test (__main__.TestMatrixDetDerivative) ... ok

----------------------------------------------------------------------
Ran 1 test in 0.418s

OK
%