4. 行列 - linalg

4. 行列

Pythonで学ぶ線形代数学

2019.10.11

問4.2

code: latex1.py

from numpy.random import randint, choice

from sympy import Matrix, latex

m, n = randint(2, 4, 2)

X = -3, -2, -1, 1, 2, 3, 4, 5

A = Matrix(choice(X, (m, n)))

B = Matrix(choice(X, (m, n)))

print(f'{latex(A)} + {latex(B)} = ')

ソースコード: latex1.py

出力例:

code: python

\left\begin{matrix}-2 & 4 & 4\\2 & -1 & 1\\-3 & 2 & 3\end{matrix}\right + \left\begin{matrix}-2 & 5 & 2\\2 & 5 & 5\\4 & 5 & 5\end{matrix}\right =

LaTeXによる出力例

https://gyazo.com/b23b1054aac38ba05dfc0a7da3d2851f

問4.3

code: random2d.py

import matplotlib.pyplot as plt

from numpy import array, random

A = array(1, 2], [2, 3)

B = array(1, 2], [2, 4)

P = random.normal(0, 1, (1000, 2))

plt.scatter(P:, 0, P:, 1, s=4)

Q = array(B.dot(p) for p in P)

plt.scatter(Q:, 0, Q:, 1, s=4, color='red')

plt.axis("equal"), plt.show()

ソースコード: random2d.py

https://gyazo.com/de7f9e69227fb98a572e925064a83a2f

code: random3d.py

from vpython import *

from numpy import array, random

A = array(1, 2, 3], 2, 3, 4, [3, 4, 5)

B = array(1, 2, 3], 2, 3, 1, [3, 1, 2)

canvas(background=color.white, foreground=color.black)

for u in 5, 0, 0], 0, 5, 0, [0, 0, 5:

v = vector(*u)

curve(pos=-v, v, color=v)

P = random.normal(0, 1, (1000, 3))

points(pos=vector(*p) for p in P, radius=4)

Q = array(A.dot(p) for p in P)

points(pos=vector(*q) for q in Q, radius=4, color=color.red)

ソースコード: random3d.py

https://gyazo.com/f4a46511c45461ddc05cb2ee0adb8fa1

回転行列

code: lena4.py

from numpy import array, pi, sin, cos

import matplotlib.pyplot as plt

t = pi / 4

A = array(cos(t), -sin(t)], [sin(t), cos(t))

with open('lena.txt', 'r') as fd:

P = eval(fd.read())

Q = A.dot(p) for p in P

x, y = zip(*Q)

plt.scatter(x, y, s=1)

plt.axis('equal'), plt.show()

ソースコード: lena4.py

https://gyazo.com/2e8f73a9b6c4616f8f45d189bea38347

問4.6

code: problems.py

from numpy import array

A = array(1, 2], [3, 4)

B = array(1, 2, 3], [4, 5, 6)

C = array(1, 2], 3, 4, [5, 6)

D = array(1, 2, 3], 4, 5, 6, [7, 8, 9)

for X in (A, B, C, D):

for Y in (A, B, C, D):

if X.shape1 == Y.shape0:

print(f'{X}\n{Y}\n= {X.dot(Y)}\n')

ソースコード: problems.py

実行結果

code: python

[1 2

3 4]

[1 2

3 4]

= [ 7 10

15 22]

[1 2

3 4]

= [ 9 12 15

[1 2

5 6]

= [22 28

= [30 36 42

[1 2

5 6]

[1 2

3 4]

= [ 7 10

[1 2

5 6]

= [ 9 12 15

[1 2

5 6]

= [ 22 28

= [ 30 36 42

code: mat_product1.py

from sympy import var, Matrix, sin, cos, eye

from sympy.abc import theta

A = Matrix([cos(theta), sin(theta),

-sin(theta), cos(theta)])

B = Matrix([1, 0,

0, -1])

C = Matrix([cos(theta), -sin(theta),

sin(theta), cos(theta)])

D = C * B * A

E = (eye(2) + D) / 2

ソースコード: mat_product1.py

code: mat_product2.py

from numpy import matrix, sin, cos, tan, pi, eye

import matplotlib.pyplot as plt

t = pi / 6

A = matrix(cos(t), sin(t)], [-sin(t), cos(t))

B = matrix(1, 0], [0, -1)

C = matrix(cos(t), -sin(t)], [sin(t), cos(t))

D = C * B * A

E = (eye(2) + D) / 2

x = matrix(5], [5)

y = D * x

z = E * x

plt.plot(0, 10, 0, 10 * tan(t))

plt.plot([x0, 0, y0, 0], [x1, 0, y1, 0])

plt.plot([x0, 0, z0, 0], [x1, 0, z1, 0])

plt.text(x0, 0, x1, 0, 'x', fontsize=18)

plt.text(y0, 0, y1, 0, 'y', fontsize=18)

plt.text(z0, 0, z1, 0, 'z', fontsize=18)

plt.axis('scaled'), plt.xlim(0, 10), plt.ylim(0, 6), plt.show()

ソースコード: mat_product2.py

https://gyazo.com/50bbb0bafff3c35e8f3b072364065ac7

問4.7

code: latex2.py

from numpy.random import randint, choice

from sympy import Matrix, latex

m, el, n = randint(2, 4, 3)

X = -3, -2, -1, 1, 2, 3, 4, 5

A = Matrix(choice(X, (m, el)))

B = Matrix(choice(X, (el, n)))

print(f'{latex(A)}{latex(B)} = ')

ソースコード: latex2.py

出力例

code: python

\left\begin{matrix}5 & -1 & 2\\-2 & 3 & 3\\2 & 3 & -3\end{matrix}\right\left\begin{matrix}3 & 5 & 2\\1 & 2 & -1\\1 & 3 & -2\end{matrix}\right =

LaTeXによる出力

https://gyazo.com/cf2534b293adb93231cd5688b3f94b19

code: mat_product3.py

from sympy import Matrix, solve, eye

from sympy.abc import a, b, c, d, e, f

A = Matrix([1, 2, 3,

2, 3, 4])

B = Matrix([a, b,

c, d,

e, f])

ans = solve(A * B - eye(2), a, b, c, d, e, f)

ソースコード: mat_product3.py

実行後の対話例

code: python

>> ans

{d: -2*f - 1, c: 2 - 2*e, b: f + 2, a: e - 3}

>> C = B.subs(ans); C

Matrix([

e - 3, f + 2,

2 - 2*e, -2*f - 1,

e, f])

>> A * C

Matrix([

1, 0,

0, 1])

>> solve(B*A - eye(3), a, b, c, d, e, f)

[]

code: mat_product4.py

def matrix_multiply(A, B):

m, el, n = len(A), len(A0), len(B0)

C = [[sum([Aik * Bkj for k in range(el)])

for j in range(n)] for i in range(m)]

return C

if __name__ == '__main__':

from numpy.random import normal

import matplotlib.pyplot as plt

from time import time

N = range(10, 210, 10)

T = []

for n in N:

A = normal(0, 1, (n, n)).tolist()

t0 = time()

matrix_multiply(A, A)

t1 = time()

print(n, end=', ')

T.append(t1 - t0)

plt.plot(N, T), plt.show()

ソースコード: mat_product4.py

https://gyazo.com/14280443bd3a5a5fcd610889f415e4f5

code: mat_product5.py

from numpy.random import normal

from numpy.linalg import inv

import matplotlib.pyplot as plt

from time import time

N = range(100, 2100, 100)

T = ], [], [

for n in N:

t0 = time()

A = normal(0, 1, (n, n))

t1 = time()

A.dot(A)

t2 = time()

inv(A)

t3 = time()

print(n, end=', ')

t = (t0, t1, t2, t3)

for i in range(3):

Ti.append(ti + 1 - ti)

label = 'f(x)', 'g(x)', 'h(x)'

for i in range(3):

plt.plot(N, Ti)

plt.text(N-1, Ti-1, labeli, fontsize=18)

plt.show()

ソースコード: mat_product5.py

https://gyazo.com/e8b67da16b028696aa3d1daa7ff0f42e