Eigenvectors of a Hermitian matrix

Question

A Hermitian matrix is a complex square matrix which is equal to its conjugate transpose. Its matrix elements fulfil following condition:

$$a_{ij} = \bar{a}_{ji}$$

Everytime, I compute eigenvectors of a Hermitian matrix using Python, the first coefficient of the eigenvector is a pure real number. Is this an attribute of Hermitian matrices?

I attach a code snippet to generate a Hermitian matrix, compute its eigenvectors and print the eigenvector corresponding to the lowest eigenvalue.

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])

This question was originally posted on StackOverflow.

Answer 1

Reposting my comment as an answer:

Any scalar multiple of any eigenvector is again an eigenvector (with the same eigenvalue), so whatever your code's output (for representing the eigenvectors) is, it could be multiplied by, e.g., $i$ and it would still be correct. So it comes down to how function calls used in your code chose to represent the eigenvectors.

See here is the source code for LA.eigh(), which is used in your code to get the eigenvectors.