特征值分解与奇异值分解(SVD)

1.使用QR分解获取特征值和特征向量

将矩阵A进行QR分解，得到正规正交矩阵Q与上三角形矩阵R。由上可知Ak为相似矩阵，当k增加时，Ak收敛到上三角矩阵，特征值为对角项。

2.奇异值分解(SVD)

 

其中U是m×m阶酉矩阵；Σ是半正定m×n阶对角矩阵；而V*，即V的共轭转置，是n×n阶酉矩阵。

将矩阵A乘它的转置，得到的方阵可用于求特征向量v，进而求出奇异值σ和左奇异向量u。

 1 #coding:utf8
 2 import numpy as np
 3 np.set_printoptions(precision=4, suppress=True)
 4 
 5 def householder_reflection(A):
 6     """Householder变换"""
 7     (r, c) = np.shape(A)
 8     Q = np.identity(r)
 9     R = np.copy(A)
10     for cnt in range(r - 1):
11         x = R[cnt:, cnt]
12         e = np.zeros_like(x)
13         e[0] = np.linalg.norm(x)
14         u = x - e
15         v = u / np.linalg.norm(u)
16         Q_cnt = np.identity(r)
17         Q_cnt[cnt:, cnt:] -= 2.0 * np.outer(v, v)
18         R = np.dot(Q_cnt, R)  # R=H(n-1)*...*H(2)*H(1)*A
19         Q = np.dot(Q, Q_cnt)  # Q=H(n-1)*...*H(2)*H(1)  H为自逆矩阵
20     return (Q, R)
21 
22 def eig(A, epsilon=1e-10):
23     '''采用QR分解法计算特征值和特征向量 '''
24     (q,r_)=householder_reflection(A)
25     h = np.identity(A.shape[0])
26     for i in range(50):
27         B=np.dot(r_,q)
28         h=h.dot(q)
29         (q,r)=gram_schmidt(B)
30         if abs(r.trace()-r_.trace())< epsilon:
31             print("Converged in {} iterations!".format(i))
32             break
33         r_=r
34     return r,h
35 
36 def svd(A):
37     '''奇异值分解'''
38     n, m = A.shape
39     svd_ = []
40     k = min(n, m)
41     v_=eig(np.dot(A.T, A))[1]  #np.linalg.eig(np.dot(A.T, A))[1]
42     for i in range(k):
43         v=v_.T[i]
44         u_ = np.dot(A, v)
45         s = np.linalg.norm(u_)
46         u = u_ / s
47         svd_.append((s, u, v))
48     ss, us, vs = [np.array(x) for x in zip(*svd_)]
49     return  us.T,ss, vs
50 
51 if __name__ == "__main__":
52 
53     mat = np.array([
54         [2, 5, 3],
55         [1, 2, 1],
56         [4, 1, 1],
57         [3, 5, 2],
58         [5, 3, 1],
59         [4, 5, 5],
60         [2, 4, 2],
61         [2, 2, 5],
62     ], dtype='float64')
63     u,s,v = svd(mat)
64     print u
65     print s
66     print v
67     print np.dot(np.dot(u,np.diag(s)),v)