


'''simple docstring'''






from __future__ import annotations

lowercase				=[]





def lowerCamelCase__     (		__lowerCamelCase			:				list[list[int]]			,       __lowerCamelCase			:				int			,       __lowerCamelCase			:				int				):







   '''simple docstring'''




   for i in range(len(__lowerCamelCase				)				):
      if board[row][i] == 1:
         return False
   for i in range(len(__lowerCamelCase				)				):
      if board[i][column] == 1:
         return False
   for i, j in zip(range(__lowerCamelCase			,       -1			,       -1				)			,       range(__lowerCamelCase			,       -1			,       -1				)				):
      if board[i][j] == 1:
         return False
   for i, j in zip(range(__lowerCamelCase			,       -1			,       -1				)			,       range(__lowerCamelCase			,       len(__lowerCamelCase				)				)				):
      if board[i][j] == 1:
         return False
   return True





def lowerCamelCase__     (		__lowerCamelCase			:				list[list[int]]			,       __lowerCamelCase			:				int				):







   '''simple docstring'''




   if row >= len(__lowerCamelCase				):
      solution.append(__lowerCamelCase				)
      printboard(__lowerCamelCase				)
      print()
      return True
   for i in range(len(__lowerCamelCase				)				):
      if is_safe(__lowerCamelCase			,       __lowerCamelCase			,       __lowerCamelCase				):
         _UpperCAmelCase  :  Optional[int]				       =1
         solve(__lowerCamelCase			,       row + 1				)
         _UpperCAmelCase  :  Optional[Any]				       =0
   return False





def lowerCamelCase__     (		__lowerCamelCase			:				list[list[int]]				):







   '''simple docstring'''




   for i in range(len(__lowerCamelCase				)				):
      for j in range(len(__lowerCamelCase				)				):
         if board[i][j] == 1:
            print('Q'			,       end=' '				)
         else:
            print('.'			,       end=' '				)
      print()


# n=int(input("The no. of queens"))
lowercase				=8
lowercase				=[[0 for i in range(n)] for j in range(n)]
solve(board, 0)




print('The total no. of solutions are :', len(solution))





'''simple docstring'''






from typing import Any

import numpy as np





def lowerCamelCase__     (		__lowerCamelCase			:				np.ndarray				):







   '''simple docstring'''




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





def lowerCamelCase__     (		__lowerCamelCase			:				np.ndarray			,       __lowerCamelCase			:				np.ndarray				):







   '''simple docstring'''




   _UpperCAmelCase  :  str				       =v.conjugate().T
   _UpperCAmelCase  :  Optional[int]				       =v_star.dot(__lowerCamelCase				)
   assert isinstance(__lowerCamelCase			,       np.ndarray				)
   return (v_star_dot.dot(__lowerCamelCase				)) / (v_star.dot(__lowerCamelCase				))





def lowerCamelCase__     (		):







   '''simple docstring'''




   _UpperCAmelCase  :  Optional[Any]				       =np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]				)
   _UpperCAmelCase  :  List[str]				       =np.array([[1], [2], [3]]				)
   assert is_hermitian(__lowerCamelCase				), f"{a} is not hermitian."
   print(rayleigh_quotient(__lowerCamelCase			,       __lowerCamelCase				)				)

   _UpperCAmelCase  :  List[str]				       =np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]				)
   assert is_hermitian(__lowerCamelCase				), f"{a} is not hermitian."
   assert rayleigh_quotient(__lowerCamelCase			,       __lowerCamelCase				) == float(3				)


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 tests()

