org.hipparchus.linear

Class ComplexEigenDecomposition

java.lang.Object

org.hipparchus.linear.ComplexEigenDecomposition

Direct Known Subclasses:

OrderedComplexEigenDecomposition

public class ComplexEigenDecomposition
extends Object

Given a matrix A, it computes a complex eigen decomposition AV = VD.

Complex Eigen Decomposition differs from the EigenDecomposition since it computes the eigen vectors as complex eigen vectors (if applicable).

Beware that in the complex case, you do not always have \(V \times V^{T} = I\) or even a diagonal matrix, even if the eigenvectors that form the columns of the V matrix are independent. On example is the square matrix \[ A = \left(\begin{matrix} 3 & -2\\ 4 & -1 \end{matrix}\right) \] which has two conjugate eigenvalues \(\lambda_1=1+2i\) and \(\lambda_2=1-2i\) with associated eigenvectors \(v_1^T = (1, 1-i)\) and \(v_2^T = (1, 1+i)\). \[ V\timesV^T = \left(\begin{matrix} 2 & 2\\ 2 & 0 \end{matrix}\right) \] which is not the identity matrix. Therefore, despite \(A \times V = V \times D\), \(A \ne V \times D \time V^T\), which would hold for real eigendecomposition.

Compute complex eigen values from the Schur transform. Compute complex eigen vectors based on eigen values and the inverse iteration method. see: https://en.wikipedia.org/wiki/Inverse_iteration https://en.wikiversity.org/wiki/Shifted_inverse_iteration http://www.robots.ox.ac.uk/~sjrob/Teaching/EngComp/ecl4.pdf http://www.math.ohiou.edu/courses/math3600/lecture16.pdf

Field Summary

Fields

Modifier and Type	Field and Description
static double	DEFAULT_EIGENVECTORS_EQUALITY Default threshold below which eigenvectors are considered equal.
static double	DEFAULT_EPSILON Default value to use for internal epsilon.
static double	DEFAULT_EPSILON_AV_VD_CHECK Internally used epsilon criteria for final AV=VD check.

Constructor Summary

Constructors

Constructor and Description
ComplexEigenDecomposition(RealMatrix matrix) Constructor for decomposition.
ComplexEigenDecomposition(RealMatrix matrix, double eigenVectorsEquality, double epsilon, double epsilonAVVDCheck) Constructor for decomposition.

Method Summary

Modifier and Type	Method and Description
protected void	checkDefinition(RealMatrix matrix) Check definition of the decomposition in runtime.
protected void	findEigenValues(RealMatrix matrix) Compute eigen values using the Schur transform.
protected void	findEigenVectors(FieldMatrix<Complex> matrix) Compute the eigen vectors using the inverse power method.
FieldMatrix<Complex>	getD() Getter D.
double	getDeterminant() Computes the determinant.
Complex[]	getEigenvalues() Getter of the eigen values.
FieldVector<Complex>	getEigenvector(int i) Getter of the eigen vectors.
FieldMatrix<Complex>	getV() Getter V.
FieldMatrix<Complex>	getVT() Getter VT.
boolean	hasComplexEigenvalues() Confirm if there are complex eigen values.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

DEFAULT_EIGENVECTORS_EQUALITY

public static final double DEFAULT_EIGENVECTORS_EQUALITY

Default threshold below which eigenvectors are considered equal.

See Also:

Constant Field Values

DEFAULT_EPSILON

public static final double DEFAULT_EPSILON

Default value to use for internal epsilon.

See Also:

Constant Field Values

DEFAULT_EPSILON_AV_VD_CHECK

public static final double DEFAULT_EPSILON_AV_VD_CHECK

Internally used epsilon criteria for final AV=VD check.

See Also:

Constant Field Values

Constructor Detail

ComplexEigenDecomposition

public ComplexEigenDecomposition(RealMatrix matrix)

Constructor for decomposition.

This constructor uses the default values DEFAULT_EIGENVECTORS_EQUALITY, DEFAULT_EPSILON and DEFAULT_EPSILON_AV_VD_CHECK

Parameters:

matrix - real matrix.

ComplexEigenDecomposition

public ComplexEigenDecomposition(RealMatrix matrix,
                                 double eigenVectorsEquality,
                                 double epsilon,
                                 double epsilonAVVDCheck)

Constructor for decomposition.

The eigenVectorsEquality threshold is used to ensure the L∞-normalized eigenvectors found using inverse iteration are different from each other. if \(min(|e_i-e_j|,|e_i+e_j|)\) is smaller than this threshold, the algorithm considers it has found again an already known vector, so it drops it and attempts a new inverse iteration with a different start vector. This value should be much larger than epsilon which is used for convergence

Parameters:

matrix - real matrix.

eigenVectorsEquality - threshold below which eigenvectors are considered equal

epsilon - Epsilon used for internal tests (e.g. is singular, eigenvalue ratio, etc.)

epsilonAVVDCheck - Epsilon criteria for final AV=VD check

Since:

1.8

Method Detail

getEigenvalues

public Complex[] getEigenvalues()

Getter of the eigen values.

Returns:

igen values.

getEigenvector

public FieldVector<Complex> getEigenvector(int i)

Getter of the eigen vectors.

Parameters:

i - which eigen vector.

Returns:

eigen vector.

hasComplexEigenvalues

public boolean hasComplexEigenvalues()

Confirm if there are complex eigen values.

Returns:

true if there are complex eigen values.

getDeterminant

public double getDeterminant()

Computes the determinant.

Returns:

the determinant.

getV

public FieldMatrix<Complex> getV()

Getter V.

Returns:

V.

getD

public FieldMatrix<Complex> getD()

Getter D.

Returns:

D.

getVT

public FieldMatrix<Complex> getVT()

Getter VT.

Returns:

VT.

findEigenValues

protected void findEigenValues(RealMatrix matrix)

Compute eigen values using the Schur transform.

Parameters:

matrix - real matrix to compute eigen values.

findEigenVectors

protected void findEigenVectors(FieldMatrix<Complex> matrix)

Compute the eigen vectors using the inverse power method.

Parameters:

matrix - real matrix to compute eigen vectors.

checkDefinition

protected void checkDefinition(RealMatrix matrix)

Check definition of the decomposition in runtime.

Parameters:

matrix - matrix to be decomposed.