厄米矩阵的特征向量[闭]

厄米矩阵是一个复方阵,它等于它的共轭转置。其矩阵元素满足以下条件:

每次,我都用Python计算厄米矩阵的特征向量,特征向量的第一个系数是一个纯实数。这是厄米矩阵的一个属性吗?

我附加了一个代码片段来生成厄米矩阵,计算其特征向量并打印对应于最低特征值的特征向量。

import numpy as np
from numpy import linalg as LA
N = 5   # Set size of a matrix
# Generate real part of the matrix at first
real_matrix = np.random.uniform(-1.0, 1.0, size=(N,N))
real_matrix = (real_matrix + real_matrix.T)/2
# Generate imaginary part of the matrix
imaginary_matrix = np.random.uniform(-1.0, 1.0, size=(N,N))
imaginary_matrix = (imaginary_matrix + imaginary_matrix.T)/2
imaginary_matrix = imaginary_matrix.astype(complex) * 1j
for row in range(N):
    for column in range(row,N):
        if row == column:
            imaginary_matrix[row][column] = 0.0
        else:
            imaginary_matrix[row][column] *= -1
# Combine real and imaginary part
matrix = real_matrix + imaginary_matrix
# Compute and print eigenvector
eigenvalues, eigenvectors = LA.eigh(matrix)
print(eigenvectors[:,0])