Gershgorin with montecarlo

 import numpy as np
from itertools import permutations

def is_in_gershgorin_disks(matrix):
    # Check if all eigenvalues of the matrix are in the Gershgorin disks
    eigenvalues = np.linalg.eigvals(matrix)
    
    n = matrix.shape[0]
    for i in range(n):
        center = matrix[i, i]
        radius = np.sum(np.abs(matrix[i, :])) - np.abs(center)
        if not np.all(np.abs(eigenvalues - center) <= radius):
            return False
    return True

def permute_rows(matrix, permutation):
    # Permute rows in the matrix based on the given permutation
    return matrix[list(permutation), :]

def monte_carlo_approximation(matrix, num_samples=1000):
    n = matrix.shape[0]
    
    for _ in range(num_samples):
        # Randomly sample a permutation
        permutation = np.random.permutation(n)
        
        # Permute rows
        permuted_matrix = permute_rows(matrix, permutation)
        
        # Check if the permuted matrix satisfies Gershgorin's Circle Theorem
        if is_in_gershgorin_disks(permuted_matrix):
            return permutation, permuted_matrix
    
    return None, None

# Input matrix
original_matrix = np.array([
    [0, 0, 1],
    [1, 0, 0],
    [0, 1, 0]
])

# Monte Carlo approximation
permutation, permuted_matrix = monte_carlo_approximation(original_matrix, num_samples=1000)

if permutation is not None:
    print(f"\nRows permutation {permutation} results in a matrix satisfying Gershgorin's Circle Theorem:")
    print(permuted_matrix)
else:
    print("No permutation found in the Monte Carlo approximation.")