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.")

Gershgorin Circle Theorem

 I want to get the matrice where all ones are next to diagonal.
Seems it could be get from 

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), :]   # Input matrix original_matrix = np.array([       [1, 0, 1, 0, 0, 0],       [0, 0, 0, 1, 0, 0],       [0, 1, 0, 0, 0, 0],       [1, 0, 0, 0, 0, 0],       [0, 0, 0, 1, 0, 1],       [0, 0, 0, 0, 1, 1] ])   # Get all possible row permutations n = original_matrix.shape[0] all_permutations = list(permutations(range(n)))   best_permutation = None best_fitness = 0.0  # Initialize with the worst fitness value   # Check each permutation for Gershgorin's Circle Theorem for idx, permutation in enumerate(all_permutations, start=1):       # Permute rows       permuted_matrix = permute_rows(original_matrix, permutation)           # Check if the permuted matrix satisfies Gershgorin's Circle Theorem       if is_in_gershgorin_disks(permuted_matrix):         # Calculate fitness value based on the sum of eigenvalues         fitness_value = np.sum(np.linalg.eigvals(permuted_matrix))                 # Update the best permutation if the current one is better         if fitness_value > best_fitness:             best_permutation = permutation             best_fitness = fitness_value           print(f"Permutation #{idx} results in a matrix satisfying Gershgorin's Circle Theorem with fitness value: {fitness_value}")       else:         print(f"Permutation #{idx} did not result in a matrix satisfying Gershgorin's Circle Theorem.")   # Print the best permutation found print("\nBest Permutation:", best_permutation)

row sorting

import numpy as np

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

num_rows = matrix.shape[0]

identity = np.eye(num_rows)

for i in range(len(matrix)):
    for j in range(len(identity)):
        product = matrix[i] * identity[j]
        if np.any(product > 0):
            print(f"{i+1} > {j+1}")