## 2019年5月28日火曜日

### 数学 - Python - 線形代数学 - 行列式 - 行列式の存在(微分、導関数と行列式、帰納法、一般化)

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)、4(行列式の存在)、練習問題10の解答を求めてみる。

1. 3×3行列式に拡張。

$\begin{array}{l}\phi \left(t\right)\\ =\mathrm{det}\left[\begin{array}{ccc}f\left(t\right)& g\left(t\right)& h\left(t\right)\\ f\text{'}\left(t\right)& g\text{'}\left(t\right)& h\text{'}\left(t\right)\\ f\text{'}\text{'}\left(t\right)& g\text{'}\text{'}\left(t\right)& h\text{'}\text{'}\left(t\right)\end{array}\right]\\ =f\left(t\right)\left(g\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)-g\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)\right)\\ -g\left(t\right)\left(f\text{'}\left(t\right)h\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\left(t\right)h\text{'}\left(t\right)\right)\\ +h\left(t\right)\left(f\text{'}\left(t\right)g\text{'}\text{'}\left(t\right)-f\text{'}\text{'}\left(t\right)g\text{'}\left(t\right)\right)\\ \phi \text{'}\left(t\right)\\ =f\left(t\right)\left(g\text{'}\left(t\right){h}^{\left(3\right)}\left(t\right)-{g}^{\left(3\right)}h\text{'}\left(t\right)\right)\\ -g\left(t\right)\left(f\text{'}\left(t\right){h}^{\left(3\right)}\left(t\right)-{f}^{\left(3\right)}\left(t\right)h\text{'}\left(t\right)\right)\\ +h\left(t\right)\left(f\text{'}\left(t\right){g}^{\left(3\right)}\left(t\right)-{f}^{\left(3\right)}\left(t\right)g\text{'}\left(t\right)\right)\\ =\mathrm{det}\left[\begin{array}{ccc}f\left(t\right)& g\left(t\right)& h\left(t\right)\\ f\text{'}\left(t\right)& g\text{'}\left(t\right)& h\text{'}\left(t\right)\\ {f}^{\left(3\right)}\left(t\right)& {g}^{\left(3\right)}\left(t\right)& {h}^{\left(3\right)}\left(t\right)\end{array}\right]\end{array}$

一般化。

$\begin{array}{l}{f}_{1}\mathrm{det}\left[\begin{array}{ccc}{f}_{2}\text{'}& \dots & {f}_{n}\text{'}\\ ⋮& & ⋮\\ {f}_{2}^{\left(n-1\right)}& \dots & {f}_{n}^{\left(n-1\right)}\end{array}\right]-\dots +{\left(-1\right)}^{n-1}{f}_{n}\mathrm{det}\left[\begin{array}{ccc}{f}_{1}\text{'}& \dots & {f}_{n-1}^{1}\\ ⋮& & ⋮\\ {f}_{1}^{\left(n-1\right)}& \dots & {f}_{n-1}^{\left(n-1\right)}\end{array}\right]\\ ={f}_{1}\mathrm{det}\left[\begin{array}{ccc}{f}_{2}\text{'}& \dots & {f}_{n}\text{'}\\ ⋮& & ⋮\\ {f}_{2}^{\left(n\right)}& \dots & {f}_{n}^{\left(n\right)}\end{array}\right]-\dots +{\left(-1\right)}^{n-1}{f}_{n}\mathrm{det}\left[\begin{array}{ccc}{f}_{1}\text{'}& \dots & {f}_{n-1}^{1}\\ ⋮& & ⋮\\ {f}_{1}^{\left(n\right)}& \dots & {f}_{n-1}^{\left(n\right)}\end{array}\right]\\ +{f}_{1}^{1}\mathrm{det}\left[\begin{array}{ccc}{f}_{2}^{1}& \dots & {f}_{n}\text{'}\\ ⋮& & ⋮\\ {f}_{2}^{\left(n-1\right)}& \dots & {f}_{n}^{\left(n-1\right)}\end{array}\right]-\dots +{\left(-1\right)}^{n-1}{f}_{n}\text{'}\mathrm{det}\left[\begin{array}{ccc}{f}_{n}\text{'}& \dots & {f}_{n-1}^{1}\\ ⋮& & ⋮\\ {f}_{1}^{\left(n-1\right)}& \dots & {f}_{n-1}^{\left(n-1\right)}\end{array}\right]\\ ={f}_{1}\mathrm{det}\left[\begin{array}{ccc}{f}_{2}\text{'}& \dots & {f}_{n}\text{'}\\ ⋮& & ⋮\\ {f}_{n}^{\left(n\right)}& \dots & {f}_{n}^{\left(n\right)}\end{array}\right]-\dots +{\left(-1\right)}^{n-1}{f}_{n}\left[\begin{array}{ccc}{f}_{1}& \dots & {f}_{n-1}^{1}\\ ⋮& & ⋮\\ {f}_{1}^{\left(n\right)}& \dots & {f}_{n-1}^{\left(n\right)}\end{array}\right]\\ =\mathrm{det}\left[\begin{array}{ccc}{f}_{1}& \dots & {f}_{n}\\ ⋮& & ⋮\\ {f}_{1}^{\left(n\right)}& \dots & {f}_{n}^{\left(n\right)}\end{array}\right]\end{array}$

よって、 帰納法により、一番下の行と微分すればし、この形の行列式の導関数が求められる。

（証明終）

コード

Python 3

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

print('10.')

t = symbols('t')

f = Function('f')(t)
g = Function('g')(t)
h = Function('h')(t)
A = Matrix([[Derivative(s, t, i) for s in [f, g, h]]
for i in range(3)])

B = Matrix([[Derivative(s, t, i) for s in [f, g, h]]
for i in [0, 1, 3]])
phi = A.det()
phi1 = B.det()
for o in [A, B, Derivative(phi, t, 1).doit() == phi1]:
pprint(o)
print()

C:\Users\...>py 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       ⎦

True

C:\Users\...>