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Create RayleighQuotient.py (#1749)

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https://en.wikipedia.org/wiki/Rayleigh_quotient

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* Update and rename RayleighQuotient.py to rayleigh_quotient.py

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* Update rayleigh_quotient.py

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* Update rayleigh_quotient.py
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"""
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()