public static void EigenvaluesAndEigenvectors()
{
SymmetricMatrix H = new SymmetricMatrix(3);
for (int r = 0; r < H.Dimension; r++)
{
for (int c = 0; c <= r; c++)
{
H[r, c] = 1.0 / (r + c + 1);
}
}
RealEigendecomposition ed = H.Eigendecomposition();
SquareMatrix V = ed.TransformMatrix;
PrintMatrix("V^T V", V.Transpose * V);
PrintMatrix("V D V^T", V * ed.DiagonalizedMatrix * V.Transpose);
foreach (RealEigenpair pair in ed.Eigenpairs)
{
Console.WriteLine($"Eigenvalue {pair.Eigenvalue}");
PrintMatrix("Hv", H * pair.Eigenvector);
PrintMatrix("ev", pair.Eigenvalue * pair.Eigenvector);
}
ed.Eigenpairs.Sort(OrderBy.MagnitudeDescending);
Console.WriteLine($"Largest eigenvalue {ed.Eigenpairs[0].Eigenvalue}");
ed.Eigenpairs.Sort(OrderBy.ValueAscending);
Console.WriteLine($"Least eigenvalue {ed.Eigenpairs[0].Eigenvalue}");
double[] eigenvalues = H.Eigenvalues();
double sum = 0.0;
double product = 1.0;
foreach (double eigenvalue in eigenvalues)
{
sum += eigenvalue;
product *= eigenvalue;
}
Console.WriteLine($"sum(e) = {sum}, tr(H) = {H.Trace()}");
Console.WriteLine($"prod(e) = {product}, det(H) = {H.CholeskyDecomposition().Determinant()}");
SquareMatrix G1 = new SquareMatrix(4);
G1[0, 3] = 1.0;
G1[1, 2] = 1.0;
G1[2, 1] = -1.0;
G1[3, 0] = -1.0;
ComplexEigendecomposition ced = G1.Eigendecomposition();
foreach (ComplexEigenpair pair in ced.Eigenpairs)
{
Console.WriteLine(pair.Eigenvalue);
}
}
public void SymmetricHilbertMatrixEigenvalues()
{
for (int d = 1; d <= 8; d++)
{
SymmetricMatrix H = TestUtilities.CreateSymmetricHilbertMatrix(d);
double tr = H.Trace();
double[] es = H.Eigenvalues();
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(es, tr));
}
}