


from typing import Any

import numpy as np


def 			UpperCAmelCase_				(      UpperCAmelCase__  ):
					return np.array_equal(UpperCAmelCase__						,							matrix.conjugate().T  )


def 			UpperCAmelCase_				(      UpperCAmelCase__						,							UpperCAmelCase__  ):
					lowercase_  							=			v.conjugate().T
					lowercase_  							=			v_star.dot(UpperCAmelCase__  )
					assert isinstance(UpperCAmelCase__						,							np.ndarray  )
					return (v_star_dot.dot(UpperCAmelCase__  )) / (v_star.dot(UpperCAmelCase__  ))


def 			UpperCAmelCase_				(      ):
					lowercase_  							=			np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]]  )
					lowercase_  							=			np.array([[1], [2], [3]]  )
					assert is_hermitian(UpperCAmelCase__  ), F'''{a} is not hermitian.'''
					print(rayleigh_quotient(UpperCAmelCase__						,							UpperCAmelCase__  )  )

					lowercase_  							=			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()






def 			UpperCAmelCase_				(      UpperCAmelCase__=2_8_1_2_3  ):
					lowercase_  							=			[1] * (limit + 1)

					for i in range(2						,							int(limit**0.5  ) + 1  ):
										sum_divs[i * i] += i
										for k in range(i + 1						,							limit // i + 1  ):
															sum_divs[k * i] += k + i

					lowercase_  							=			set()
					lowercase_  							=			0

					for n in range(1						,							limit + 1  ):
										if sum_divs[n] > n:
															abundants.add(UpperCAmelCase__  )

										if not any((n - a in abundants) for a in abundants  ):
															res += n

					return res


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


