复厄米矩阵Lanczos算法特征向量与numpy.linalg.eig结果不一致问题

我需要求解复值厄米（Hermitian）矩阵的最小特征值与对应特征向量。针对实值厄米矩阵，如下代码运行完全正常：

import numpy as np
import numpy.linalg as la
import copy as cp

# Generates the representation in Krylov subspace of a Hermitian NxN matrix using the Lanczos algorithm and an initial vector guess vg.
def Lanczos(H,vg):
    Lv=np.zeros((len(vg),len(vg)), dtype=complex) # Creates matrix for Lanczos vectors
    Hk=np.zeros((len(vg),len(vg)), dtype=complex) # Creates matrix for the Hamiltonian in Krylov subspace
    Lv[0]=vg/la.norm(vg) # Creates the first Lanczos vector as the normalized guess vector vg
     
    # Performs the first iteration step of the Lanczos algorithm
    w=np.dot(H,Lv[0]) 
    a=np.dot(np.conj(w),Lv[0])
    w=w-a*Lv[0]
    Hk[0,0]=a
     
    # Performs the iterative steps of the Lanczos algorithm
    for j in range(1,len(vg)):
        b=(np.dot(np.conj(w),np.transpose(w)))**0.5
        Lv[j]=w/b
         
        w=np.dot(H,Lv[j])
        a=np.dot(np.conj(w),Lv[j])
        w=w-a*Lv[j]-b*Lv[j-1]
        
        # Creates tridiagonal matrix Hk using a and b values
        Hk[j,j]=a
        Hk[j-1,j]=b
        Hk[j,j-1]=np.conj(b)
        
    return (Hk,Lv)

但针对复值矩阵，输出的特征值一致，特征向量却不匹配：

H = np.random.rand(8,8) + np.random.rand(8,8)*1j # Generates random complex-valued 8x8 matrix
H = H + H.conj().T # Ensures 8x8 matrix is symmetric (hermitian)
Hc = cp.copy(H)
a,b = np.linalg.eig(H) # Directly computes eigenvalues of H
vg = np.random.rand(8) + np.random.rand(8)*1j # Generates random guess vector
Hk,Lv = Lanczos(Hc,vg) # Applies the Lanczos algorithm to H using the guess vector vg
A,B= np.linalg.eig(Hk)

# While the following outputs are the same
a[0]
A[0]
# The following are not!!!
b[:,0]
np.dot(B[:,0],Lv)

请问我的实现存在什么问题？

解决方案 #

感谢Acccumulation的提示，该计算流程实际没有错误，特征值验证结果如下：

a[0]*b[:,0] - np.dot(H,b[:,0])
array([ 3.55271368e-15+1.11022302e-15j,  4.44089210e-16-3.65257395e-16j,
        0.00000000e+00+9.99200722e-16j, -4.44089210e-16+3.33066907e-16j,
       -2.22044605e-15-1.66533454e-16j,  1.33226763e-15+0.00000000e+00j,
        2.66453526e-15+1.22124533e-15j,  3.10862447e-15+1.22124533e-15j])
A[0]*np.dot(B[:,0],Lv) - np.dot(H,)
array([2.22044605e-15+1.77635684e-15j, 2.66453526e-15+1.55431223e-15j,
       2.66453526e-15+1.55431223e-15j, 2.66453526e-15+1.77635684e-15j,
       3.55271368e-15+2.66453526e-15j, 1.77635684e-15+1.99840144e-15j,
       1.33226763e-15+1.77635684e-15j, 8.88178420e-16+1.55431223e-15j])

该结果说明b[:,0]和np.dot(B[:,0],Lv)均为厄米矩阵H的合法特征向量。

内容的提问来源于stack exchange，提问作者ALKochListon