Python使用NumPy进行矩阵计算的案例详解

import numpy as np

# 从列表创建矩阵
matrix_2d = np.array([[1,2], [3,4]])
# 创建单位矩阵
identity = np.eye(3)
# 创建零矩阵和全一矩阵
zeros = np.zeros((2,3))
ones = np.ones((3,2))

A = np.array([[1,2], [3,4]])
B = np.array([[5,6], [7,8]])

# 对应元素相加
print(A + B)  # [[6,8], [10,12]]
# 对应元素相乘
print(A * B)  # [[5,12], [21,32]]

# 标准矩阵乘法
C = np.dot(A, B)
# 或使用@运算符
D = A @ B
print(D)  # [[19,22], [43,50]]

# 转置
transposed = A.T
# 逆矩阵
inv_A = np.linalg.inv(A)
print(np.round(A @ inv_A, 2))  # 验证单位矩阵

matrix = np.array([[1,2,3], [4,5,6]])
vector = np.array([10, 20, 30])
result = matrix + vector  # 每行加vector → [[11,22,33], [14,25,36]]

# 多维索引
matrix = np.array([[1,2,3], [4,5,6], [7,8,9]])
print(matrix[1,2])  # 输出6
print(matrix[:, 1])  # 输出[2,5,8]
# 布尔索引
mask = matrix > 4
print(matrix[mask])  # 输出[5,6,7,8,9]

# 计算特征值和特征向量
eigenvalues, eigenvectors = np.linalg.eig(A)
print(eigenvalues)  # 输出特征值
print(eigenvectors)  # 输出特征向量

X = np.array([[1,2], [3,4], [5,6]])
# 计算线性核矩阵
K = X @ X.T

from PIL import Image
img = Image.open('image.jpg').convert('L')
img_array = np.array(img)
# 3x3均值滤波
kernel = np.ones((3,3))/9
filtered = cv2.filter2D(img_array, -1, kernel)

returns = np.array([[0.01,0.02], [0.03,0.01], [0.02,0.015]])
cov_matrix = np.cov(returns, rowvar=False)
portfolio_variance = np.dot(weights.T, np.dot(cov_matrix, weights))

# 低效实现
result = np.zeros((1000,1000))
for i in range(1000):
    for j in range(1000):
        result[i,j] = A[i,j] * B[i,j]

# 高效向量化实现
optimized = A * B  # 速度提升100倍以上

# 使用float32减少内存占用
data = np.random.rand(1000,1000).astype(np.float32)
# 大数据集内存映射
large_data = np.memmap('huge_data.dat', dtype='float32', mode='r', shape=(10000,10000))

# 使用BLAS加速矩阵乘法
from scipy.linalg import blas
result = blas.gemm(A, B)

try:
    A @ B
except ValueError as e:
    print(f"矩阵维度不匹配: {e}")

from scipy.linalg import pinv
# 计算伪逆矩阵
pseudo_inv = np.linalg.pinv(A)