






'''simple docstring'''

from typing import Any

import numpy as np







def 					lowercase__				(   __UpperCamelCase					)->				Dict:
      return np.array_equal(lowercase__	,		matrix.conjugate().T					)







def 					lowercase__				(   __UpperCamelCase	,		__UpperCamelCase					)->				Any:
      UpperCamelCase       						=      v.conjugate().T
      UpperCamelCase       						=      v_star.dot(lowercase__					)
      assert isinstance(lowercase__	,		np.ndarray					)
      return (v_star_dot.dot(lowercase__					)) / (v_star.dot(lowercase__					))







def 					lowercase__				(   )->				Tuple:
      UpperCamelCase       						=      np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]]					)
      UpperCamelCase       						=      np.array([[1], [2], [3]]					)
      assert is_hermitian(lowercase__					), F"{a} is not hermitian."
      print(rayleigh_quotient(lowercase__	,		lowercase__					)					)

      UpperCamelCase       						=      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 					lowercase__				(   __UpperCamelCase	,		__UpperCamelCase					)->				str:
      if not isinstance(__UpperCamelCase	,		__UpperCamelCase					):
            raise ValueError("""iterations must be defined as integers"""					)
      if not isinstance(__UpperCamelCase	,		__UpperCamelCase					) or not number >= 1:
            raise ValueError(
                """starting number must be
                         and integer and be more than 0"""					)
      if not iterations >= 1:
            raise ValueError("""Iterations must be done more than 0 times to play FizzBuzz"""					)

      UpperCamelCase       						=      """"""
      while number <= iterations:
            if number % 3 == 0:
                  out += "Fizz"
            if number % 5 == 0:
                  out += "Buzz"
            if 0 not in (number % 3, number % 5):
                  out += str(__UpperCamelCase					)

            # print(out)
            number += 1
            out += " "
      return out


if __name__ == "__main__":
     import doctest

     doctest.testmod()
