线性代数：向量与矩阵

import numpy as np

# 创建向量
v1 = np.array([3, 4])
v2 = np.array([1, -2])

print(f"v1 = {v1}")
print(f"v2 = {v2}")
print(f"维度: {v1.shape}")

# 向量空间的基
e1 = np.array([1, 0, 0])  # x 轴单位向量
e2 = np.array([0, 1, 0])  # y 轴单位向量
e3 = np.array([0, 0, 1])  # z 轴单位向量

# 任意三维向量可表示为基的线性组合
v = 3*e1 + 4*e2 + 5*e3
print(f"线性组合结果: {v}")

import numpy as np

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])

# 向量加减
print(f"a + b = {a + b}")   # [5 7 9]
print(f"a - b = {a - b}")   # [-3 -3 -3]

# 点积（内积）
dot = np.dot(a, b)
print(f"a · b = {dot}")     # 32

# 叉积
cross = np.cross(a, b)
print(f"a × b = {cross}")

# 夹角计算（通过点积）
cos_theta = dot / (np.linalg.norm(a) * np.linalg.norm(b))
theta = np.arccos(cos_theta)
print(f"夹角: {np.degrees(theta):.2f} 度")

# 余弦相似度
similarity = dot / (np.linalg.norm(a) * np.linalg.norm(b))
print(f"余弦相似度: {similarity:.4f}")

import numpy as np

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

# 矩阵加法
print(f"A + B =
{A + B}")

# 矩阵转置
print(f"A 的转置 =
{A.T}")

# 单位矩阵
I = np.eye(2)
print(f"单位矩阵 =
{I}")

# 逐元素乘法（Hadamard 积）
print(f"A * B =
{A * B}")

# 标量乘法
print(f"3 * A =
{3 * A}")

# 矩阵形状
print(f"A 的形状: {A.shape}")  # (2, 2)
print(f"A 的行数: {A.shape[0]}")
print(f"A 的列数: {A.shape[1]}")

import numpy as np

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

# 矩阵乘法
C = np.matmul(A, B)
# 或 C = A @ B
print(f"A @ B =
{C}")

# 验证不满足交换律
print(f"B @ A =
{B @ A}")
print(f"A@B == B@A: {np.array_equal(A@B, B@A)}")  # False

# 矩阵作为线性变换：旋转 90 度
theta = np.pi / 2
R = np.array([
    [np.cos(theta), -np.sin(theta)],
    [np.sin(theta),  np.cos(theta)]
])

v = np.array([1, 0])
v_rotated = R @ v
print(f"原向量: {v}")
print(f"旋转后: {v_rotated}")  # 近似 [0, 1]

import numpy as np

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

# 行列式
det = np.linalg.det(A)
print(f"det(A) = {det:.2f}")  # -2.00

# 逆矩阵
A_inv = np.linalg.inv(A)
print(f"A 的逆 =
{A_inv}")

# 验证 A × A_inv = I
print(f"A @ A_inv =
{np.round(A @ A_inv, 10)}")

# 解线性方程组 Ax = b
b = np.array([5, 11])
x = np.linalg.solve(A, b)
print(f"解 x = {x}")  # [1. 2.]

# 验证
print(f"A @ x = {A @ x}")  # [ 5. 11.]

# 奇异矩阵（不可逆）
S = np.array([[1, 2], [2, 4]])
print(f"det(S) = {np.linalg.det(S):.2f}")  # 0.00，不可逆

import numpy as np

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

# 特征值分解
eigenvalues, eigenvectors = np.linalg.eig(A)
print(f"特征值: {eigenvalues}")
print(f"特征向量:
{eigenvectors}")

# 验证 Av = λv
v = eigenvectors[:, 0]
lam = eigenvalues[0]
print(f"A @ v = {A @ v}")
print(f"λ * v = {lam * v}")

# 特征值分解重构矩阵
D = np.diag(eigenvalues)  # 特征值对角矩阵
P = eigenvectors            # 特征向量矩阵

# A = P D P^{-1}
A_reconstructed = P @ D @ np.linalg.inv(P)
print(f"重构矩阵:
{np.round(A_reconstructed, 10)}")

# 迹 = 特征值之和
print(f"迹: {np.trace(A)} = {sum(eigenvalues)}")
# 行列式 = 特征值之积
print(f"det: {np.linalg.det(A):.2f} = {np.prod(eigenvalues):.2f}")

import numpy as np

# 综合演练：向量与矩阵操作
# 1. 创建各种向量和矩阵
v = np.array([1, 2, 3])
M = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])

# 2. 广播机制
print(f"v + 10 = {v + 10}")         # 标量广播
print(f"M + v =
{M + v}")          # 向量逐行广播

# 3. 范数计算
print(f"||v|| = {np.linalg.norm(v):.2f}")  # L2 范数
print(f"||v||_1 = {np.linalg.norm(v, 1):.2f}")  # L1 范数

# 4. 完整的特征分解实战
# 创建对称正定矩阵（确保实数特征值）
A = np.array([[4, 2], [2, 3]])
eigenvalues, eigenvectors = np.linalg.eig(A)

print(f"特征值: {eigenvalues}")
print(f"特征向量:
{eigenvectors}")

# 5. PCA 降维示例
X = np.random.randn(100, 5)  # 100 个样本，5 个特征
cov_matrix = np.cov(X.T)
eigenvalues, eigenvectors = np.linalg.eigh(cov_matrix)

# 取最大的两个特征值对应的特征向量
top_2 = eigenvectors[:, -2:]
X_pca = X @ top_2
print(f"PCA 降维后形状: {X_pca.shape}")  # (100, 2)