Create RayleighQuotient.py (#1749)

* Create RayleighQuotient.py

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

* Update RayleighQuotient.py

* Update and rename RayleighQuotient.py to rayleigh_quotient.py

* Update rayleigh_quotient.py

* Update rayleigh_quotient.py

python/black

* Update rayleigh_quotient.py

linear_algebra/src/rayleigh_quotient.py

@@ -0,0 +1,65 @@
+"""
+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()