

"""simple docstring"""






from typing import Any

import numpy as np



def   lowercase_	(    _lowercase       :		np.ndarray	):




					'''simple docstring'''






					return np.array_equal(_lowercase							,      matrix.conjugate().T	)



def   lowercase_	(    _lowercase       :		np.ndarray							,      _lowercase       :		np.ndarray	):




					'''simple docstring'''






					UpperCAmelCase			:   str      =				v.conjugate().T
					UpperCAmelCase			:   Tuple      =				v_star.dot(_lowercase	)
					assert isinstance(_lowercase							,      np.ndarray	)
					return (v_star_dot.dot(_lowercase	)) / (v_star.dot(_lowercase	))



def   lowercase_	(    ):




					'''simple docstring'''






					UpperCAmelCase			:   Optional[int]      =				np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]	)
					UpperCAmelCase			:   List[str]      =				np.array([[1], [2], [3]]	)
					assert is_hermitian(_lowercase	), F"""{a} is not hermitian."""
					print(rayleigh_quotient(_lowercase							,      _lowercase	)	)

					UpperCAmelCase			:   Dict      =				np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]	)
					assert is_hermitian(_lowercase	), F"""{a} is not hermitian."""
					assert rayleigh_quotient(_lowercase							,      _lowercase	) == float(3	)


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	tests()





"""simple docstring"""






from __future__ import annotations

snake_case_   :   str        =       list[list[int]]

# assigning initial values to the grid
snake_case_   :   Matrix        =       [
    [3, 0, 6, 5, 0, 8, 4, 0, 0],
    [5, 2, 0, 0, 0, 0, 0, 0, 0],
    [0, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
snake_case_   :   Matrix        =       [
    [5, 0, 6, 5, 0, 8, 4, 0, 3],
    [5, 2, 0, 0, 0, 0, 0, 0, 2],
    [1, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]



def   lowercase_	(    _lowercase       :		Matrix							,      _lowercase       :		int							,      _lowercase       :		int							,      _lowercase       :		int	):




					'''simple docstring'''






					for i in range(9	):
										if grid[row][i] == n or grid[i][column] == n:
															return False

					for i in range(3	):
										for j in range(3	):
															if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
																				return False

					return True



def   lowercase_	(    _lowercase       :		Matrix	):




					'''simple docstring'''






					for i in range(9	):
										for j in range(9	):
															if grid[i][j] == 0:
																				return i, j
					return None



def   lowercase_	(    _lowercase       :		Matrix	):




					'''simple docstring'''






					if location := find_empty_location(_lowercase	):
										UpperCAmelCase      ,							UpperCAmelCase			:   List[str]      =				location
					else:
										# If the location is ``None``, then the grid is solved.
										return grid

					for digit in range(1							,      10	):
										if is_safe(_lowercase							,      _lowercase							,      _lowercase							,      _lowercase	):
															UpperCAmelCase			:   int      =				digit

															if sudoku(_lowercase	) is not None:
																				return grid

															UpperCAmelCase			:   Optional[Any]      =				0

					return None



def   lowercase_	(    _lowercase       :		Matrix	):




					'''simple docstring'''






					for row in grid:
										for cell in row:
															print(_lowercase							,      end=" "	)
										print()


if __name__ == "__main__":
	# make a copy of grid so that you can compare with the unmodified grid
	for example_grid in (initial_grid, no_solution):
		print("""\nExample grid:\n""" + """=""" * 2_0)
		print_solution(example_grid)
		print("""\nExample grid solution:""")
		snake_case_   :   Dict        =       sudoku(example_grid)
		if solution is not None:
			print_solution(solution)
		else:
			print("""Cannot find a solution.""")


