





from typing import Any

import numpy as np





def 						lowerCamelCase__     (     UpperCamelCase__			:       np.ndarray						)			-> bool:



					'''simple docstring'''




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





def 						lowerCamelCase__     (     UpperCamelCase__			:       np.ndarray       ,    UpperCamelCase__			:       np.ndarray						)			-> Any:



					'''simple docstring'''




					_snake_case									=				v.conjugate().T
					_snake_case									=				v_star.dot(UpperCamelCase__						)
					assert isinstance(UpperCamelCase__       ,    np.ndarray						)
					return (v_star_dot.dot(UpperCamelCase__						)) / (v_star.dot(UpperCamelCase__						))





def 						lowerCamelCase__     (     )			-> None:



					'''simple docstring'''




					_snake_case									=				np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]						)
					_snake_case									=				np.array([[1], [2], [3]]						)
					assert is_hermitian(UpperCamelCase__						), F'''{a} is not hermitian.'''
					print(rayleigh_quotient(UpperCamelCase__       ,    UpperCamelCase__						)						)

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


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					tests()










import random





def 						lowerCamelCase__     (     UpperCamelCase__			:       list       ,    UpperCamelCase__			:       List[Any]						)			-> tuple:



					'''simple docstring'''




					_snake_case							,     _snake_case							,     _snake_case									=				[], [], []
					for element in data:
										if element < pivot:
															less.append(UpperCamelCase__						)
										elif element > pivot:
															greater.append(UpperCamelCase__						)
										else:
															equal.append(UpperCamelCase__						)
					return less, equal, greater





def 						lowerCamelCase__     (     UpperCamelCase__			:       list       ,    UpperCamelCase__			:       int						)			-> Dict:



					'''simple docstring'''




					if index >= len(UpperCamelCase__						) or index < 0:
										return None

					_snake_case									=				items[random.randint(0       ,    len(UpperCamelCase__						) - 1						)]
					_snake_case									=				0
					_snake_case							,     _snake_case							,     _snake_case									=				_partition(UpperCamelCase__       ,    UpperCamelCase__						)
					_snake_case									=				len(UpperCamelCase__						)
					_snake_case									=				len(UpperCamelCase__						)

					# index is the pivot
					if m <= index < m + count:
										return pivot
					# must be in smaller
					elif m > index:
										return quick_select(UpperCamelCase__       ,    UpperCamelCase__						)
					# must be in larger
					else:
										return quick_select(UpperCamelCase__       ,    index - (m + count)						)



