/// <summary>
/// Householder Reflection: Evaluate symmetric? unitary Q such that Q*v = -sigma*||v||*e1
/// </summary>
public static ComplexMatrix Reflection(ComplexVector v)
{
Complex sigma = v[0].Sign;
ComplexVector u = v.Clone();
u[0] += sigma * v.Norm();
ComplexMatrix m = ComplexMatrix.Identity(v.Length, v.Length);
m.MultiplyAccumulateInplace(u.TensorMultiply(u), -2d / u.ScalarMultiply(u));
return(m);
}
public void TestOrthogonalReflectionComplex()
{
StandardDistribution gaussian = new StandardDistribution();
for (int i = 0; i < 100; i++)
{
ComplexVector v = new ComplexVector(new Complex[] { Complex.Random(gaussian), Complex.Random(gaussian), Complex.Random(gaussian), Complex.Random(gaussian) });
ComplexMatrix reflection = Orthogonal.Reflection(v);
ComplexVector res = reflection * v;
Assert.That(reflection.HermitianTranspose() * reflection, NumericIs.AlmostEqualTo(ComplexMatrix.Identity(4, 4)), "orthogonal reflection matrix");
Assert.That(reflection[0, 0].IsReal, "c1 real");
Assert.That(reflection[1, 1].IsReal, "c2 real");
Assert.That(reflection[2, 2].IsReal, "c3 real");
Assert.That(reflection[3, 3].IsReal, "c4 real");
Assert.That(res[0].Modulus, NumericIs.AlmostEqualTo(v.Norm(), 1e-12), "res(1)");
Assert.That(res[1], NumericIs.AlmostEqualTo((Complex)0, 1e-12), "res(2)");
Assert.That(res[2], NumericIs.AlmostEqualTo((Complex)0, 1e-12), "res(3)");
Assert.That(res[3], NumericIs.AlmostEqualTo((Complex)0, 1e-12), "res(4)");
}
}