eigenvectors from eigenvalues

vecs_from_vals.py

# To calculate eigenvectors of a Hermitian matrix using nothing but the eigenvalues
# from https://arxiv.org/pdf/1908.03795.pdf
import numpy as np

# use numpy to calculate the eigen values
e_vals = np.linalg.eigvals(mat)
    
eigen_vectors = np.zeros_like(mat)

n, _ = mat.shape
for i in range(n):
    for j in range(n):
        lhs_product = 1
        for k in range(n):
            if k != i:
                lhs_product *= (e_vals[i] - e_vals[k])

        # remove the jth row and column from the matrix
        Mj = np.delete(np.delete(mat, j, axis=0), j, axis=1)
        e_vals_j = np.linalg.eigvals(Mj)
        
        rhs_product = 1
        for k in range(n-1):
            rhs_product *= (e_vals[i] - e_vals_j[k])
    
        eigen_vectors[i,j] = np.sqrt(rhs_product / lhs_product)