2019年12月20日金曜日


ラング線形代数学(上)
原著
楽天ブックス Yahoo!

学習環境

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1章(R^nにおけるベクトル)、6(複素数)、練習問題6の解答を求めてみる。



    1. ⟨ A , B ⟩ = ∑ k = 1 n α k β - k = ∑ k = 1 n β - k α k = ∑ k = 1 n β k α - k - = ⟨ B , A ⟩ -

    2. ⟨ A , B + C ⟩ = ∑ k = 1 n α k β k + γ k - = ∑ k = 1 n α k β - k + γ - k = ∑ k = 1 n α k β - k + α k γ - k = ∑ k = 1 n α k β - k + ∑ k = 1 n α k γ - k = ⟨ A , B ⟩ + ⟨ A , C ⟩

    3. ⟨ α A , B ⟩ = ∑ k = 1 n α α k β - k = α ∑ k = 1 n α k β - k = α ⟨ A , B ⟩ ⟨ A , α B ⟩ = ∑ k = 1 n α k α β k - = ∑ k = 1 n α k α - β - k = α - ∑ k = 1 n α k β - k = α - ⟨ A , B ⟩

    4. ⟨ A , A ⟩ = ∑ k = 1 n α k α - k = ∑ k = 1 n α k 2 ≥ 0

      よって、

      A = O

      ならば

      k = 1 , … , n α k = 0 ⟨ A , A ⟩ = O

      また、

      A ≠ O

      ならば、

      α l ≠ 0

      となる l が存在するから

      ⟨ A , A ⟩ > 0

コード

#!/usr/bin/env python3
from unittest import TestCase, main
from sympy import symbols, Matrix, I

print('6')

n = 2
A = Matrix([symbols([f'a{i}' for i in range(1, n + 1)], imag=True)])
B = Matrix([symbols([f'b{i}' for i in range(1, n + 1)], imag=True)])
C = Matrix([symbols([f'c{i}' for i in range(1, n + 1)], imag=True)])
alpha = symbols('α', imag=True)


def dot(a, b):
    return sum([a0 * b0.conjugate() for a0, b0 in zip(a, b)])


class MyTestCase(TestCase):
    def test_sp1(self):
        self.assertEqual(dot(A, B), dot(B, A).conjugate())

    def test_sp2(self):
        self.assertEqual(dot(A, B + C).expand(),
                         (dot(A, B) + dot(A, C)).expand())

    def test_sp3_1(self):
        self.assertEqual(dot(alpha * A, B).expand(),
                         (alpha * dot(A, B)).expand())

    def test_sp3_2(self):
        self.assertEqual(dot(A, alpha * B).expand(),
                         (alpha.conjugate() * dot(A, B)).expand())

    def test_sp4_1(self):
        O = Matrix([0 for _ in range(n)])
        self.assertEqual(dot(O, O), 0)

    def test_sp4_2(self):
        A = Matrix([1, 0, 0, 0, 0])
        B = Matrix([I, 0, 0, 0, 0])
        C = Matrix([1 - I, 0, 0, 0, 0])
        for t in [A, B, C]:
            self.assertGreater(dot(A, A), 0)


if __name__ == '__main__':
    main()

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

% ./sample6.py -v
6
test_sp1 (__main__.MyTestCase) ... ok
test_sp2 (__main__.MyTestCase) ... ok
test_sp3_1 (__main__.MyTestCase) ... ok
test_sp3_2 (__main__.MyTestCase) ... ok
test_sp4_1 (__main__.MyTestCase) ... ok
test_sp4_2 (__main__.MyTestCase) ... ok

----------------------------------------------------------------------
Ran 6 tests in 0.058s

OK
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時刻: 13:00 ラベル: , , , ,

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