import numpy as np
# 創(chuàng)建一維數(shù)組
arr1 = np.array([1, 2, 3, 4])
# 創(chuàng)建二維數(shù)組（矩陣）
matrix = np.array([[1, 2, 3], [4, 5, 6]])
# 指定數(shù)據(jù)類(lèi)型
arr_float = np.array([1, 2, 3], dtype=np.float64)
print("一維數(shù)組:", arr1)
print("二維矩陣:\n", matrix)
print("數(shù)據(jù)類(lèi)型:", arr_float.dtype)

# 創(chuàng)建矩陣對(duì)象（不推薦新代碼使用）
mat = np.matrix([[1, 2], [3, 4]])
print("矩陣對(duì)象:\n", mat)
print("類(lèi)型:", type(mat))

# 創(chuàng)建3x3全零矩陣
zeros_3x3 = np.zeros((3, 3))
# 創(chuàng)建2x3x4全零三維數(shù)組
zeros_3d = np.zeros((2, 3, 4))

print("3x3零矩陣:\n", zeros_3x3)
print("2x3x4零數(shù)組形狀:", zeros_3d.shape)

# 創(chuàng)建2x4全1矩陣
ones_2x4 = np.ones((2, 4))
# 創(chuàng)建帶數(shù)據(jù)類(lèi)型的全1矩陣
ones_int = np.ones((3, 3), dtype=int)
print("2x4全1矩陣:\n", ones_2x4)

# 創(chuàng)建3x3單位矩陣
I3 = np.eye(3)
# 創(chuàng)建4x4單位矩陣
I4 = np.identity(4)
# 創(chuàng)建非方陣的對(duì)角矩陣
rect_eye = np.eye(3, 5)
print("3x3單位矩陣:\n", I3)
print("3x5對(duì)角矩陣:\n", rect_eye)

# 從對(duì)角線創(chuàng)建矩陣
diag_matrix = np.diag([1, 2, 3, 4])
print("對(duì)角矩陣:\n", diag_matrix)

# 從矩陣提取對(duì)角線
matrix = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
diagonal = np.diag(matrix)
print("矩陣對(duì)角線:", diagonal)

A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
# 三種矩陣乘法方式
result1 = np.dot(A, B)
result2 = A @ B  # Python 3.5+ 推薦
result3 = A.dot(B)
print("A @ B = \n", result1)
print("結(jié)果是否相同:", np.array_equal(result1, result2))

# 二維矩陣乘法
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
result = np.matmul(A, B)
# 批量矩陣乘法（三維）
batch_A = np.random.randn(10, 3, 4)  # 10個(gè)3x4矩陣
batch_B = np.random.randn(10, 4, 5)  # 10個(gè)4x5矩陣
batch_result = np.matmul(batch_A, batch_B)  # 得到10個(gè)3x5矩陣
print("單個(gè)矩陣乘法:\n", result)
print("批量矩陣乘法形狀:", batch_result.shape)

# 向量?jī)?nèi)積
v1 = np.array([1, 2, 3])
v2 = np.array([4, 5, 6])
inner_product = np.inner(v1, v2)  # 1*4 + 2*5 + 3*6 = 32
print("向量?jī)?nèi)積:", inner_product)

v1 = np.array([1, 2, 3])
v2 = np.array([4, 5, 6, 7])
outer_product = np.outer(v1, v2)

print("外積矩陣:\n", outer_product)
print("形狀:", outer_product.shape)

# 3D向量叉積
a = np.array([1, 0, 0])
b = np.array([0, 1, 0])
cross_product = np.cross(a, b)  # [0, 0, 1]
print("叉積:", cross_product)

A = np.array([[1, 2], [3, 4]])
try:
    A_inv = np.linalg.inv(A)
    print("矩陣A:\n", A)
    print("逆矩陣:\n", A_inv)
    print("驗(yàn)證 A @ A_inv:\n", A @ A_inv)  # 應(yīng)接近單位矩陣
except np.linalg.LinAlgError:
    print("矩陣不可逆")

# 對(duì)于非方陣或奇異矩陣
A = np.array([[1, 2], [3, 4], [5, 6]])
A_pinv = np.linalg.pinv(A)
print("矩陣A:\n", A)
print("偽逆矩陣:\n", A_pinv)
print("A @ A_pinv @ A ≈ A:\n", A @ A_pinv @ A)

A = np.array([[1, 2], [3, 4]])
det_A = np.linalg.det(A)
print("矩陣A:\n", A)
print("行列式:", det_A)

A = np.array([[1, -1], [2, 4]])
eigenvalues, eigenvectors = np.linalg.eig(A)
print("矩陣A:\n", A)
print("特征值:", eigenvalues)
print("特征向量:\n", eigenvectors)
# 驗(yàn)證：A * v = λ * v
for i in range(len(eigenvalues)):
    v = eigenvectors[:, i]
    λ = eigenvalues[i]
    print(f"驗(yàn)證特征向量 {i}: A*v = {A @ v}, λ*v = {λ * v}")

A = np.array([[1, 2], [3, 4], [5, 6]])
U, S, Vh = np.linalg.svd(A, full_matrices=False)

print("矩陣A:\n", A)
print("U矩陣:\n", U)
print("奇異值:", S)
print("Vh矩陣:\n", Vh)

# 重建矩陣
Sigma = np.diag(S)
A_reconstructed = U @ Sigma @ Vh
print("重建矩陣:\n", A_reconstructed)

A = np.array([[1, 2], [3, 4], [5, 6]])
Q, R = np.linalg.qr(A)

print("矩陣A:\n", A)
print("Q矩陣:\n", Q)
print("R矩陣:\n", R)
print("驗(yàn)證 Q*R:\n", Q @ R)

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

print("矩陣:\n", A)
print("所有元素和:", np.sum(A))
print("每列和:", np.sum(A, axis=0))
print("每行和:", np.sum(A, axis=1))

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

print("矩陣:\n", A)
print("全局均值:", np.mean(A))
print("列均值:", np.mean(A, axis=0))
print("行均值:", np.mean(A, axis=1))

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

# 兩種轉(zhuǎn)置方式
A_transpose1 = np.transpose(A)
A_transpose2 = A.T

print("原始矩陣:\n", A)
print("轉(zhuǎn)置矩陣:\n", A_transpose1)
print("兩種方法結(jié)果相同:", np.array_equal(A_transpose1, A_transpose2))

# 解方程組:
# 1*x + 2*y = 5
# 3*x + 4*y = 6
A = np.array([[1, 2], [3, 4]])
b = np.array([5, 6])
x = np.linalg.solve(A, b)

print("系數(shù)矩陣 A:\n", A)
print("常數(shù)項(xiàng) b:", b)
print("解 x:", x)
print("驗(yàn)證 A*x = b:", A @ x)

# 擬合數(shù)據(jù)點(diǎn) y = 2x + 1 + 噪聲
x = np.array([0, 1, 2, 3, 4])
y = np.array([1, 3, 5, 7, 9]) + np.random.normal(0, 0.2, 5)
# 構(gòu)建設(shè)計(jì)矩陣 A = [[1, x1], [1, x2], ...]
A = np.vstack([np.ones_like(x), x]).T
# 求解最小二乘問(wèn)題
coeff, residuals, rank, s = np.linalg.lstsq(A, y, rcond=None)
print(f"擬合直線: y = {coeff[1]:.3f}x + {coeff[0]:.3f}")
print(f"殘差平方和: {residuals[0]:.3f}")

A = np.arange(12)  # 0到11
A_reshaped = A.reshape(3, 4)

print("原始數(shù)組:", A)
print("重塑為3x4:\n", A_reshaped)

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

# 垂直連接（行方向）
C_vertical = np.concatenate((A, B), axis=0)
# 水平連接（列方向）
C_horizontal = np.concatenate((A, B), axis=1)

print("矩陣A:\n", A)
print("矩陣B:\n", B)
print("垂直連接:\n", C_vertical)
print("水平連接:\n", C_horizontal)

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

# 垂直堆疊
C_vertical = np.vstack((A, B))
# 水平堆疊
C_horizontal = np.hstack((A, B))

print("A:", A)
print("B:", B)
print("垂直堆疊:\n", C_vertical)
print("水平堆疊:", C_horizontal)

import matplotlib.pyplot as plt
import numpy as np

# 創(chuàng)建可視化示例
fig, axes = plt.subplots(2, 3, figsize=(12, 8))

# 1. 單位矩陣可視化
I = np.eye(5)
axes[0, 0].imshow(I, cmap='Blues', interpolation='nearest')
axes[0, 0].set_title('單位矩陣')
axes[0, 0].set_xticks(range(5))
axes[0, 0].set_yticks(range(5))

# 2. 隨機(jī)矩陣可視化
random_matrix = np.random.randn(5, 5)
axes[0, 1].imshow(random_matrix, cmap='RdYlBu', interpolation='nearest')
axes[0, 1].set_title('隨機(jī)矩陣')
axes[0, 1].set_xticks(range(5))
axes[0, 1].set_yticks(range(5))

# 3. 矩陣乘法示例
A = np.array([[1, 0.5], [0.5, 1]])
B = np.array([[1, 0], [0, 0.5]])
C = A @ B
axes[0, 2].imshow(C, cmap='viridis', interpolation='nearest')
axes[0, 2].set_title('矩陣乘法結(jié)果')
axes[0, 2].set_xticks(range(2))
axes[0, 2].set_yticks(range(2))

# 4. 特征向量可視化
eig_matrix = np.array([[2, 1], [1, 2]])
eigvals, eigvecs = np.linalg.eig(eig_matrix)
axes[1, 0].quiver([0, 0], [0, 0], eigvecs[0, :], eigvecs[1, :], 
                  angles='xy', scale_units='xy', scale=1, color=['r', 'b'])
axes[1, 0].set_xlim(-1.5, 1.5)
axes[1, 0].set_ylim(-1.5, 1.5)
axes[1, 0].set_title('特征向量')
axes[1, 0].grid(True)
axes[1, 0].set_aspect('equal')

# 5. SVD分解可視化
U, S, Vh = np.linalg.svd(eig_matrix)
axes[1, 1].imshow(U @ np.diag(S) @ Vh, cmap='plasma', interpolation='nearest')
axes[1, 1].set_title('SVD重建矩陣')
axes[1, 1].set_xticks(range(2))
axes[1, 1].set_yticks(range(2))

# 6. 3D矩陣可視化（使用2D切片）
matrix_3d = np.random.rand(4, 4, 4)
axes[1, 2].imshow(matrix_3d[:, :, 0], cmap='coolwarm', interpolation='nearest')
axes[1, 2].set_title('3D矩陣的2D切片')
axes[1, 2].set_xticks(range(4))
axes[1, 2].set_yticks(range(4))

plt.tight_layout()
plt.show()

# 性能優(yōu)化示例
A = np.random.randn(1000, 1000)
B = np.random.randn(1000, 1000)
# 更快的矩陣乘法（使用優(yōu)化的BLAS庫(kù)）
result = A @ B  # 自動(dòng)使用優(yōu)化的實(shí)現(xiàn)
# 避免不必要的復(fù)制
C = np.empty_like(A)
np.matmul(A, B, out=C)  # 指定輸出位置