





from typing import Any

import numpy as np



def 							lowerCAmelCase_      (				A_):
    return np.array_equal(_snake_case  ,matrix.conjugate().T)



def 							lowerCAmelCase_      (				A_  ,A_):
    UpperCamelCase__:     List[str]     							=			v.conjugate().T
    UpperCamelCase__:     str     							=			v_star.dot(_snake_case)
    assert isinstance(_snake_case  ,np.ndarray)
    return (v_star_dot.dot(_snake_case)) / (v_star.dot(_snake_case))



def 							lowerCAmelCase_      (				):
    UpperCamelCase__:     List[Any]     							=			np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]])
    UpperCamelCase__:     Dict     							=			np.array([[1], [2], [3]])
    assert is_hermitian(_snake_case), F"{a} is not hermitian."
    print(rayleigh_quotient(_snake_case  ,_snake_case))

    UpperCamelCase__:     Any     							=			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()



"""simple docstring"""




def lowercase_  (   _snake_case						):
							if a < 0:
														raise ValueError("""Input value must be a positive integer"""						)
							elif isinstance(_snake_case   ,_snake_case						):
														raise TypeError("""Input value must be a 'int' type"""						)
							return bin(_snake_case						).count("""1"""						)


if __name__ == "__main__":
					import doctest

					doctest.testmod()




