Python/linear_algebra/src/rayleigh_quotient.py

"""
https://en.wikipedia.org/wiki/Rayleigh_quotient
"""
import numpy as np


def is_hermitian(matrix:np.matrix) -> bool:
    """
    Checks if a matrix is Hermitian.

    >>> import numpy as np
    >>> A = np.matrix([
    ... [2,    2+1j, 4],
    ... [2-1j,  3,  1j],
    ... [4,    -1j,  1]])
    >>> is_hermitian(A)
    True
    >>> A = np.matrix([
    ... [2,    2+1j, 4+1j],
    ... [2-1j,  3,  1j],
    ... [4,    -1j,  1]])
    >>> is_hermitian(A)
    False
    """
    return np.array_equal(matrix, matrix.H)


def rayleigh_quotient(A:np.matrix, v:np.matrix) -> float:
    """
    Returns the Rayleigh quotient of a Hermitian matrix A and
    vector v.
    >>> import numpy as np
    >>> A = np.matrix([
    ... [1,  2, 4],
    ... [2,  3,  -1],
    ... [4, -1,  1]
    ... ])
    >>> v = np.matrix([
    ... [1],
    ... [2],
    ... [3]
    ... ])
    >>> rayleigh_quotient(A, v)
    matrix([[3.]])
    """
    v_star = v.H
    return (v_star * A * v) / (v_star * v)


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

    A = np.matrix([[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()