矩阵特征向量与奇异值分解

calendar Apr 14, 2026 · 2 分钟阅读 · 矩阵 特征向量 奇异值分解 线性代数  ·
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特征向量-直方图

1from scipy.linalg import eig  # 矩阵分解
2import numpy as np   # 生成矩阵
3import matplotlib.pyplot as plt  # 绘图

1A = [[1, 2],#生成一个2*2的矩阵
2     [2, 1]] 
3A

[[1, 2], [2, 1]]

1
2# 特征分解
3evals, evecs = eig(A) #求A的特征值（evals）和特征向量(evecs)

[[ 0.70710678 -0.70710678]
 [ 0.70710678  0.70710678]]

1evecs = evecs[:, 0], evecs[:, 1]
2print(evecs)

(array([0.70710678, 0.70710678]), array([-0.70710678,  0.70710678]))

 1fig, ax = plt.subplots()
 2for spine in ['left', 'bottom']:#让在左下角的坐标轴经过原点
 3    ax.spines[spine].set_position('zero')
 4    ax.grid(alpha=0.4)
 5xmin, xmax = -3, 3
 6ymin, ymax = -3, 3
 7ax.set(xlim=(xmin, xmax), ylim=(ymin, ymax))
 8for v in evecs:
 9    ax.annotate(s="", xy=v, xytext=(0, 0),
10                arrowprops=dict(facecolor='blue',
11                shrink=0,
12                alpha=0.6,
13                width=0.5))
14x = np.linspace(xmin, xmax, 3)#在指定的间隔内返回均匀间隔的数字
15for v in evecs:
16    a = v[1] / v[0] #沿特征向量方向的单位向量
17    ax.plot(x, a * x, 'r-', lw=0.4)# 参数 lw 表示图线的粗细
18plt.show()

png

奇异值分解

1import numpy as np
2import matplotlib.pyplot as plt

1words = ["books","dad","stock","value","singular","estate","decomposition"]

1X=np.array([[0,2,1,0,0,0,0],[2,0,0,1,0,1,0],[1,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,1,0,1],[0,1,0,0,1,0,0],[0,0,0,0,1,1,1]])

1X

array([[0, 2, 1, 0, 0, 0, 0],
       [2, 0, 0, 1, 0, 1, 0],
       [1, 0, 0, 0, 0, 0, 1],
       [0, 0, 1, 0, 0, 0, 0],
       [0, 1, 0, 0, 0, 0, 0],
       [0, 0, 0, 1, 1, 0, 1],
       [0, 1, 0, 0, 1, 0, 0],
       [0, 0, 0, 0, 1, 1, 1]])

1U,s,Vh=np.linalg.svd(X)

1print("U=",U)

U= [[-1.87135757e-01 -7.93624528e-01  2.45011855e-01 -2.05404352e-01
  -3.88578059e-16  5.75779114e-16 -2.57394431e-01 -4.08248290e-01]
 [-6.92896814e-01  2.88368077e-01  5.67788037e-01  2.22142537e-01
   2.54000254e-01 -6.37019839e-16 -2.21623012e-02  2.05865892e-17]
 [-3.53233681e-01  1.22606651e-01  3.49203461e-02 -4.51735990e-01
  -7.62000762e-01  1.27403968e-15  2.72513448e-01  3.80488702e-17]
 [-2.61369658e-02 -1.33189110e-01  7.51079037e-02 -6.44727454e-01
   5.08000508e-01  1.77635684e-15  3.68146235e-01  4.08248290e-01]
 [-8.04993957e-02 -3.30217709e-01  8.49519758e-02  2.19661551e-01
  -2.54000254e-01 -4.81127681e-16 -3.12770333e-01  8.16496581e-01]
 [-3.95029694e-01  1.56123876e-02 -5.28290830e-01 -6.82340484e-02
   1.27000127e-01 -7.07106781e-01 -2.09360158e-01  1.55512464e-17]
 [-2.02089013e-01 -3.80395849e-01 -2.12899198e-01  4.80790894e-01
   8.04483689e-16 -1.60632798e-15  7.33466480e-01  1.76241226e-16]
 [-3.95029694e-01  1.56123876e-02 -5.28290830e-01 -6.82340484e-02
   1.27000127e-01  7.07106781e-01 -2.09360158e-01 -1.23226632e-16]]

1print("s=",s)

s= [2.85653844 2.63792139 2.06449303 1.14829917 1.         1.
 0.54848559]

1print("Vh",Vh)

Vh [[-6.08788345e-01 -2.29949618e-01 -7.46612474e-02 -3.80854846e-01
  -3.47325416e-01 -3.80854846e-01 -4.00237243e-01]
 [ 2.65111314e-01 -8.71088358e-01 -3.51342402e-01  1.15234846e-01
  -1.32365989e-01  1.15234846e-01  5.83153945e-02]
 [ 5.66965547e-01  1.75382762e-01  1.55059743e-01  1.91316736e-02
  -6.14911671e-01  1.91316736e-02 -4.94872736e-01]
 [-6.48865369e-03  2.52237176e-01 -7.40339999e-01  1.34031699e-01
   2.99854608e-01  1.34031699e-01 -5.12239408e-01]
 [-2.54000254e-01 -2.54000254e-01  5.08000508e-01  3.81000381e-01
   2.54000254e-01  3.81000381e-01 -5.08000508e-01]
 [ 0.00000000e+00 -7.68640544e-16  2.33583082e-15 -7.07106781e-01
  -1.21802199e-15  7.07106781e-01  1.91457709e-15]
 [ 4.16034348e-01 -1.71550021e-01  2.01922906e-01 -4.22112199e-01
   5.73845817e-01 -4.22112199e-01 -2.66564648e-01]]

1plt.axis([-0.8,0.2,-0.8,0.8])
2for i in range(len(words)):
3    plt.text(U[i,0],U[i,1],words[i])
4plt.show()

png