#### Rayleigh Quotient

p
```"""
https://en.wikipedia.org/wiki/Rayleigh_quotient
"""

from typing import Any

import numpy as np

def is_hermitian(matrix: np.ndarray) -> bool:
"""
Checks if a matrix is Hermitian.
>>> import numpy as np
>>> A = np.array([
... [2,    2+1j, 4],
... [2-1j,  3,  1j],
... [4,    -1j,  1]])
>>> is_hermitian(A)
True
>>> A = np.array([
... [2,    2+1j, 4+1j],
... [2-1j,  3,  1j],
... [4,    -1j,  1]])
>>> is_hermitian(A)
False
"""
return np.array_equal(matrix, matrix.conjugate().T)

def rayleigh_quotient(a: np.ndarray, v: np.ndarray) -> Any:
"""
Returns the Rayleigh quotient of a Hermitian matrix A and
vector v.
>>> import numpy as np
>>> A = np.array([
... [1,  2, 4],
... [2,  3,  -1],
... [4, -1,  1]
... ])
>>> v = np.array([
... [1],
... [2],
... [3]
... ])
>>> rayleigh_quotient(A, v)
array([[3.]])
"""
v_star = v.conjugate().T
v_star_dot = v_star.dot(a)
assert isinstance(v_star_dot, np.ndarray)
return (v_star_dot.dot(v)) / (v_star.dot(v))

def tests() -> None:
a = np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]])
v = np.array([[1], [2], [3]])
assert is_hermitian(a), f"{a} is not hermitian."
print(rayleigh_quotient(a, v))

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

if __name__ == "__main__":
import doctest

doctest.testmod()
tests()
```