

from typing import Any

import numpy as np





def      snake_case_				(__A					:						np.ndarray					)      ->     bool:
   return np.array_equal(__A ,       matrix.conjugate().T					)





def      snake_case_				(__A					:						np.ndarray ,       __A					:						np.ndarray					)      ->     Any:
   __lowerCAmelCase  :      Optional[int]    =						v.conjugate().T
   __lowerCAmelCase  :      Optional[int]    =						v_star.dot(__A					)
   assert isinstance(__A ,       np.ndarray					)
   return (v_star_dot.dot(__A					)) / (v_star.dot(__A					))





def      snake_case_				()      ->     None:
   __lowerCAmelCase  :      Optional[int]    =						np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]]					)
   __lowerCAmelCase  :      int    =						np.array([[1], [2], [3]]					)
   assert is_hermitian(__A					), f'''{a} is not hermitian.'''
   print(rayleigh_quotient(__A ,       __A					)					)

   __lowerCAmelCase  :      Union[str, Any]    =						np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]					)
   assert is_hermitian(__A					), f'''{a} is not hermitian.'''
   assert rayleigh_quotient(__A ,       __A					) == float(3					)


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   tests()





def      snake_case_				()      ->     List[Any]:
   for n in range(1 ,       1_0_0_0_0_0_0					):
      yield n * (n + 1) // 2





def      snake_case_				(__A					:						Dict					)      ->     Tuple:
   __lowerCAmelCase  :      Optional[int]    =						1
   __lowerCAmelCase  :      Optional[int]    =						2
   while i * i <= n:
      __lowerCAmelCase  :      Optional[int]    =						0
      while n % i == 0:
         n //= i
         multiplicity += 1
      divisors_count *= multiplicity + 1
      i += 1
   if n > 1:
      divisors_count *= 2
   return divisors_count





def      snake_case_				()      ->     Dict:
   return next(i for i in triangle_number_generator() if count_divisors(__A					) > 5_0_0					)


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


