
from typing import Any

import numpy as np




def 	A_ (							snake_case							:   np.ndarray )      ->    bool:



						'''simple docstring'''

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




def 	A_ (							snake_case							:   np.ndarray ,      snake_case							:   np.ndarray )      ->    Any:



						'''simple docstring'''

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




def 	A_ (							)      ->    None:



						'''simple docstring'''

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

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


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				tests()






def 	A_ (							snake_case							:   float )      ->    float:



						'''simple docstring'''

						if edge <= 0 or not isinstance(snake_case ,      snake_case ):
												raise ValueError('''Length must be a positive.''' )
						return 3 * ((25 + 10 * (5 ** (1 / 2))) ** (1 / 2)) * (edge**2)




def 	A_ (							snake_case							:   float )      ->    float:



						'''simple docstring'''

						if edge <= 0 or not isinstance(snake_case ,      snake_case ):
												raise ValueError('''Length must be a positive.''' )
						return ((15 + (7 * (5 ** (1 / 2)))) / 4) * (edge**3)


if __name__ == "__main__":
				import doctest

				doctest.testmod()




