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70 lines
1.5 KiB
Python
70 lines
1.5 KiB
Python
"""
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https://en.wikipedia.org/wiki/Rayleigh_quotient
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"""
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from typing import Any
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import numpy as np
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def is_hermitian(matrix: np.ndarray) -> bool:
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"""
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Checks if a matrix is Hermitian.
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>>> import numpy as np
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>>> A = np.array([
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... [2, 2+1j, 4],
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... [2-1j, 3, 1j],
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... [4, -1j, 1]])
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>>> is_hermitian(A)
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True
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>>> A = np.array([
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... [2, 2+1j, 4+1j],
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... [2-1j, 3, 1j],
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... [4, -1j, 1]])
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>>> is_hermitian(A)
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False
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"""
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return np.array_equal(matrix, matrix.conjugate().T)
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def rayleigh_quotient(a: np.ndarray, v: np.ndarray) -> Any:
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"""
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Returns the Rayleigh quotient of a Hermitian matrix A and
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vector v.
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>>> import numpy as np
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>>> A = np.array([
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... [1, 2, 4],
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... [2, 3, -1],
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... [4, -1, 1]
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... ])
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>>> v = np.array([
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... [1],
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... [2],
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... [3]
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... ])
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>>> rayleigh_quotient(A, v)
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array([[3.]])
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"""
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v_star = v.conjugate().T
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v_star_dot = v_star.dot(a)
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assert isinstance(v_star_dot, np.ndarray)
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return (v_star_dot.dot(v)) / (v_star.dot(v))
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def tests() -> None:
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a = np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]])
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v = np.array([[1], [2], [3]])
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assert is_hermitian(a), f"{a} is not hermitian."
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print(rayleigh_quotient(a, v))
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a = np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]])
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assert is_hermitian(a), f"{a} is not hermitian."
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assert rayleigh_quotient(a, v) == float(3)
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if __name__ == "__main__":
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import doctest
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doctest.testmod()
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tests()
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