rayleigh quotient Algorithm
If we repair the complex matrix M, then the resulting Rayleigh quotient map (considered as a function of X) completely determines M via the polarization identity; indeed, this remains true even if we let M to be non-Hermitian. In quantum mechanics, the Rayleigh quotient gives the expectation value of the observable corresponding to the operator M for a system whose state is given by x.
"""
https://en.wikipedia.org/wiki/Rayleigh_quotient
"""
import numpy as np
def is_hermitian(matrix: np.array) -> 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.array, v: np.array) -> float:
"""
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
return (v_star.dot(A).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()
