



from typing import Any

import numpy as np

def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			np.ndarray    ):
				"""simple docstring"""







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

def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			np.ndarray				,     _SCREAMING_SNAKE_CASE       :			np.ndarray    ):
				"""simple docstring"""







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

def   lowerCAmelCase__						(							):
				"""simple docstring"""







				__a       =		np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]    )
				__a       =		np.array([[1], [2], [3]]    )
				assert is_hermitian(_SCREAMING_SNAKE_CASE    ), f"{a} is not hermitian."
				print(rayleigh_quotient(_SCREAMING_SNAKE_CASE				,     _SCREAMING_SNAKE_CASE    )    )

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


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						tests()





from __future__ import annotations

from collections import Counter
from random import random






class    SCREAMING_SNAKE_CASE			:





							def __init__(   self		:		Optional[Any]	):

											'''simple docstring'''
											__a       =		{}





							def 						UpperCamelCase_					(   self		:		Dict      ,				__lowercase		:		str	):

											'''simple docstring'''
											__a       =		{}





							def 						UpperCamelCase_					(   self		:		List[str]      ,				__lowercase		:		str      ,				__lowercase		:		str      ,				__lowercase		:		float	):

											'''simple docstring'''
											if nodea not in self.connections:
															self.add_node(__lowercase	)
											if nodea not in self.connections:
															self.add_node(__lowercase	)
											__a       =		probability





							def 						UpperCamelCase_					(   self		:		Union[str, Any]	):

											'''simple docstring'''
											return list(self.connections	)







							def 						UpperCamelCase_					(   self		:		Tuple      ,				__lowercase		:		str	):

											'''simple docstring'''
											__a       =		0
											__a       =		random()

											for dest in self.connections[node]:
															current_probability += self.connections[node][dest]
															if current_probability > random_value:
																			return dest
											return ""



def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			str				,     _SCREAMING_SNAKE_CASE       :			list[tuple[str, str, float]]				,     _SCREAMING_SNAKE_CASE       :			int    ):
				"""simple docstring"""







				__a       =		MarkovChainGraphUndirectedUnweighted()

				for nodea, nodea, probability in transitions:
								graph.add_transition_probability(_SCREAMING_SNAKE_CASE				,     _SCREAMING_SNAKE_CASE				,     _SCREAMING_SNAKE_CASE    )

				__a       =		Counter(graph.get_nodes()    )
				__a       =		start

				for _ in range(_SCREAMING_SNAKE_CASE    ):
								__a       =		graph.transition(_SCREAMING_SNAKE_CASE    )
								visited[node] += 1

				return visited


if __name__ == "__main__":
						import doctest

						doctest.testmod()
