


'''simple docstring'''

from typing import Any

import numpy as np





def lowerCamelCase      (      lowerCAmelCase    :     np.ndarray		):





	"""simple docstring"""







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





def lowerCamelCase      (      lowerCAmelCase    :     np.ndarray		,		lowerCAmelCase    :     np.ndarray		):





	"""simple docstring"""







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





def lowerCamelCase      (      ):





	"""simple docstring"""







	__magic_name__	:   Dict					     =					np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]]		)
	__magic_name__	:   Any					     =					np.array([[1], [2], [3]]		)
	assert is_hermitian(lowercase_		), f'{a} is not hermitian.'
	print(rayleigh_quotient(lowercase_		,		lowercase_		)		)

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


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							tests()



'''simple docstring'''

def lowerCamelCase      (      lowerCAmelCase    :     str		,		lowerCAmelCase    :     Any		,		lowerCAmelCase    :     Any=False		):





	"""simple docstring"""







	if isinstance(lowerCAmelCase		,		lowerCAmelCase		) and isinstance(lowerCAmelCase		,		lowerCAmelCase		):
		__magic_name__	:   str					     =					len(set_a.intersection(lowerCAmelCase		)		)

		if alternative_union:
			__magic_name__	:   List[str]					     =					len(lowerCAmelCase		) + len(lowerCAmelCase		)
		else:
			__magic_name__	:   Any					     =					len(set_a.union(lowerCAmelCase		)		)

		return intersection / union

	if isinstance(lowerCAmelCase		,		(list, tuple)		) and isinstance(lowerCAmelCase		,		(list, tuple)		):
		__magic_name__	:   str					     =					[element for element in set_a if element in set_b]

		if alternative_union:
			__magic_name__	:   Dict					     =					len(lowerCAmelCase		) + len(lowerCAmelCase		)
			return len(lowerCAmelCase		) / union
		else:
			__magic_name__	:   Any					     =					set_a + [element for element in set_b if element not in set_a]
			return len(lowerCAmelCase		) / len(lowerCAmelCase		)

		return len(lowerCAmelCase		) / len(lowerCAmelCase		)
	return None


if __name__ == "__main__":
							lowerCAmelCase				:Dict        =      {'''a''', '''b''', '''c''', '''d''', '''e'''}
							lowerCAmelCase				:Tuple        =      {'''c''', '''d''', '''e''', '''f''', '''h''', '''i'''}
							print(jaccard_similarity(set_a, set_b))