public void SymmetricRandomMatrixEigenvectors()
{
for (int d = 1; d <= 100; d = d + 11)
{
Console.WriteLine("d={0}", d);
SymmetricMatrix M = CreateSymmetricRandomMatrix(d, 1);
double tr = M.Trace();
RealEigensystem E = M.Eigensystem();
Assert.IsTrue(E.Dimension == M.Dimension);
SquareMatrix D = new SquareMatrix(E.Dimension);
double[] es = new double[E.Dimension];
for (int i = 0; i < E.Dimension; i++)
{
double e = E.Eigenvalue(i);
ColumnVector v = E.Eigenvector(i);
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(M, v, e));
D[i, i] = e;
es[i] = e;
}
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(es, tr));
Assert.IsTrue(TestUtilities.IsNearlyEqual(
D, E.TransformMatrix.Transpose() * M * E.TransformMatrix
));
}
}
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));
}
}
public void SymmetricRandomMatrixEigenvectors()
{
for (int d = 1; d <= 100; d = d + 11)
{
SymmetricMatrix M = CreateSymmetricRandomMatrix(d, 1);
RealEigendecomposition E = M.Eigendecomposition();
Assert.IsTrue(E.Dimension == M.Dimension);
double[] eigenvalues = new double[E.Dimension];
for (int i = 0; i < E.Dimension; i++)
{
double e = E.Eigenpairs[i].Eigenvalue;
ColumnVector v = E.Eigenpairs[i].Eigenvector;
// The eigenvector works
Assert.IsTrue(TestUtilities.IsNearlyEigenpair(M, v, e));
// The eigenvalue is the corresponding diagonal value of D
Assert.IsTrue(E.DiagonalizedMatrix[i, i] == e);
// Remember eigenvalue to take sum in future
eigenvalues[i] = e;
}
// The eigenvectors sum to trace
double tr = M.Trace();
Assert.IsTrue(TestUtilities.IsSumNearlyEqual(eigenvalues, tr));
// The decomposition works
Assert.IsTrue(TestUtilities.IsNearlyEqual(
E.DiagonalizedMatrix, E.TransformMatrix.Transpose * M * E.TransformMatrix
));
// Transform matrix is orthogonal
Assert.IsTrue(TestUtilities.IsNearlyEqual(
E.TransformMatrix.Transpose * E.TransformMatrix, UnitMatrix.OfDimension(d)
));
}
}
