Matrix in Python

Libraries ライブラリ
Matrix generation 行列の生成
Transpose matrix 転置行列
Diagonal matrix 対角行列
Trace トレース
Determinant 行列式
Inverse matrix 逆行列
Scalar multiplication スカラー倍
Matrix equality 行列の相等
Addition, Subtraction 加算、減算
Dot product 内積
System of linear equations 連立一次方程式
EigenValue, EigenVector 固有値、固有ベクトル
Reference 参考

Libraries ライブラリ

import numpy as np

Matrix generation 行列の生成

x = np.array([[1, 0], \
              [0, 1]])
print(type(x))
# <class 'numpy.ndarray'>
print(x)
# [[1 0]      
#  [0 1]]

x = np.array([[1, 2], \
              [3, 4], \
              [5, 6]])
print(x.shape)  # Number of rows and columns
# (3, 2)
print(x[2, 0])  # Element
# 5

x = np.array([[1, 2, 3], \
              [4, 5, 6], \
              [7, 8, 9]])
print(x[:, 1])  # Column
# [2 5 8]
print(x[1, :])  # Row
# [4 5 6]

Zero matrix ゼロ行列

print(np.zeros((2, 3)))
# [[0. 0. 0.] 
#  [0. 0. 0.]]

Matrix of all ones 全ての要素が1の行列

print(np.ones((2, 3)))
# [[1. 1. 1.] 
#  [1. 1. 1.]]

Identity matrix 単位行列

print(np.eye(3))
# [[1. 0. 0.] 
#  [0. 1. 0.] 
#  [0. 0. 1.]]

Transpose matrix 転置行列

x = np.array([[1, 2], \
              [3, 4]])
print(x.T)  # Transpose
# [[1 3]
#  [2 4]]

Diagonal matrix 対角行列

x = np.array([[1, 2], \
              [3, 4]])
print(np.diagonal(x))  # Diagonal
# [1 4]

Trace トレース

x = np.array([[1, 2, 3], \
              [4, 5, 6], \
              [7, 8, 9]])
print(np.trace(x))  # Trace
# 15

Determinant 行列式

theta = (1/4) * np.pi
r = np.array([[np.cos(theta), -np.sin(theta)], \
              [np.sin(theta), np.cos(theta)]])
print(np.linalg.det(r)) # Determinant
# 1.0

Inverse matrix 逆行列

theta = (1/4) * np.pi
r = np.array([[np.cos(theta), -np.sin(theta)], \
              [np.sin(theta), np.cos(theta)]])
print(np.linalg.inv(r)) # Inverse
# [[ 0.70710678  0.70710678]
#  [-0.70710678  0.70710678]]

Scalar multiplication スカラー倍

x = np.array([[1, 2], \
              [3, 4]])
print(2 * x)  # Scalar multiplication
# [[2 4]
#  [6 8]]

Matrix equality 行列の相等

i = np.eye(2)
iinv = np.linalg.inv(i)
print(np.array_equal(i, iinv))
# True

theta = (1/4) * np.pi
r = np.array([[np.cos(theta), -np.sin(theta)], \
              [np.sin(theta), np.cos(theta)]])
rinv = np.linalg.inv(r)
print(np.array_equal(r, rinv))
# False

Addition, Subtraction 加算、減算

i = np.eye(2)
x = np.array([[1, 2], \
              [3, 4]])
print(x + i)  # Addition
# [[2. 2.]
#  [3. 5.]]
print(x - i)  # Subtraction
# [[0. 2.]
#  [3. 3.]]

Dot product 内積

i = np.eye(2)
x = np.array([[1, 2], \
              [3, 4]])
prod = np.dot(x, i) # Dot product
print(prod)
# [[1. 2.]
#  [3. 4.]]

theta = (1/4) * np.pi
r = np.array([[np.cos(theta), -np.sin(theta)], \
              [np.sin(theta), np.cos(theta)]])
rinv = np.linalg.inv(r)
prod = np.dot(r, rinv)
print(prod)
# [[1.00000000e+00 1.01465364e-17]
#  [1.01465364e-17 1.00000000e+00]]

System of linear equations 連立一次方程式

a = np.array([[1, 1], [1, -1]])
b = np.array([3, 1])
x = np.linalg.solve(a, b) # Solve
print(x)
# [2. 1.]

EigenValue, EigenVector 固有値、固有ベクトル

a = np.array([[3, -1], [1, 1]])
eigenvalues, eigenvectors = np.linalg.eig(a)  # EigenValue, EigenVector
print(eigenvalues)
# [2.00000002 1.99999998]
print(eigenvectors)
# [[0.70710679 0.70710677]
#  [0.70710677 0.70710679]]