


from typing import Any

import numpy as np
def 				UpperCAmelCase_					(							__snake_case						)     ->			bool:







 """simple docstring"""


 return np.array_equal(__snake_case     ,						matrix.conjugate().T						)
def 				UpperCAmelCase_					(							__snake_case     ,						__snake_case						)     ->			Any:







 """simple docstring"""


 _lowercase							      =v.conjugate().T
 _lowercase							      =v_star.dot(__snake_case						)
 assert isinstance(__snake_case     ,						np.ndarray						)
 return (v_star_dot.dot(__snake_case						)) / (v_star.dot(__snake_case						))
def 				UpperCAmelCase_					(							)     ->			None:







 """simple docstring"""


 _lowercase							      =np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]]						)
 _lowercase							      =np.array([[1], [2], [3]]						)
 assert is_hermitian(__snake_case						), F"{a} is not hermitian."
 print(rayleigh_quotient(__snake_case     ,						__snake_case						)						)

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


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  tests()


"""simple docstring"""






def SCREAMING_SNAKE_CASE__			(   __UpperCAmelCase=2_8_1_2_3 )  ->  Any:
   lowercase__:   Optional[Any]					=      [1] * (limit + 1)

   for i in range(2       ,    int(limit**0.5 ) + 1 ):
      sum_divs[i * i] += i
      for k in range(i + 1       ,    limit // i + 1 ):
         sum_divs[k * i] += k + i

   lowercase__:   Union[str, Any]					=      set()
   lowercase__:   Optional[Any]					=      0

   for n in range(1       ,    limit + 1 ):
      if sum_divs[n] > n:
         abundants.add(__UpperCAmelCase )

      if not any((n - a in abundants) for a in abundants ):
         res += n

   return res


if __name__ == "__main__":
       print(solution())
