




from typing import Any

import numpy as np





def        __UpperCAmelCase						(							lowerCamelCase_ :     np.ndarray  )       ->  bool:







			"""simple docstring"""





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





def        __UpperCAmelCase						(							lowerCamelCase_ :     np.ndarray  ,							lowerCamelCase_ :     np.ndarray  )       ->  Any:







			"""simple docstring"""





			SCREAMING_SNAKE_CASE_   : Optional[Any]					  =      v.conjugate().T
			SCREAMING_SNAKE_CASE_   : Any					  =      v_star.dot(lowerCamelCase_  )
			assert isinstance(lowerCamelCase_  ,							np.ndarray  )
			return (v_star_dot.dot(lowerCamelCase_  )) / (v_star.dot(lowerCamelCase_  ))





def        __UpperCAmelCase						(							)       ->  None:







			"""simple docstring"""





			SCREAMING_SNAKE_CASE_   : List[Any]					  =      np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]  )
			SCREAMING_SNAKE_CASE_   : List[str]					  =      np.array([[1], [2], [3]]  )
			assert is_hermitian(lowerCamelCase_  ), F'{a} is not hermitian.'
			print(rayleigh_quotient(lowerCamelCase_  ,							lowerCamelCase_  )  )

			SCREAMING_SNAKE_CASE_   : Any					  =      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()






from math import sqrt





def    __lowerCamelCase      (					UpperCamelCase__ ):







							'''simple docstring'''






							if 1 < number < 4:
														# 2 and 3 are primes
														return True
							elif number < 2 or number % 2 == 0 or number % 3 == 0:
														# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
														return False

							# All primes number are in format of 6k +/- 1
							for i in range(5  ,				int(sqrt(UpperCamelCase__ ) + 1 )  ,				6 ):
														if number % i == 0 or number % (i + 2) == 0:
																					return False
							return True





def    __lowerCamelCase      (					UpperCamelCase__ = 10001 ):







							'''simple docstring'''






							snake_case_     							=       0
							snake_case_     							=       1
							while count != nth and number < 3:
														number += 1
														if is_prime(UpperCamelCase__ ):
																					count += 1
							while count != nth:
														number += 2
														if is_prime(UpperCamelCase__ ):
																					count += 1
							return number


if __name__ == "__main__":
					print(F'''{solution() = }''')




