Eigenvalues and eigenvectors of a real matrix. If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal. I.e. A = V.Multiply(D.Multiply(V.Transpose())) and V.Multiply(V.Transpose()) equals the identity matrix. If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of V represent the eigenvectors in the sense that A*V = V*D, i.e. A.Multiply(V) equals V.Multiply(D). The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*Inverse(V) depends upon V.cond(). More...

Inheritance diagram for DotNetMatrix.EigenvalueDecomposition:

Collaboration diagram for DotNetMatrix.EigenvalueDecomposition:

Public Member Functions
	EigenvalueDecomposition (GeneralMatrix Arg)
	Check for symmetry, then construct the eigenvalue decomposition More...

virtual GeneralMatrix	GetV ()
	Return the eigenvector matrix More...

	EigenvalueDecomposition (GeneralMatrix Arg)
	Check for symmetry, then construct the eigenvalue decomposition More...

virtual GeneralMatrix	GetV ()
	Return the eigenvector matrix More...

Properties
virtual double[]	RealEigenvalues `[get]`
	Return the real parts of the eigenvalues More...

virtual double[]	ImagEigenvalues `[get]`
	Return the imaginary parts of the eigenvalues More...

virtual GeneralMatrix	D `[get]`
	Return the block diagonal eigenvalue matrix More...

Private Member Functions
void	tred2 ()

void	tql2 ()

void	orthes ()

void	cdiv (double xr, double xi, double yr, double yi)

void	hqr2 ()

void ISerializable.	GetObjectData (SerializationInfo info, StreamingContext context)

void	tred2 ()

void	tql2 ()

void	orthes ()

void	cdiv (double xr, double xi, double yr, double yi)

void	hqr2 ()

void ISerializable.	GetObjectData (SerializationInfo info, StreamingContext context)

Private Attributes
int	n
	Row and column dimension (square matrix). matrix dimension. More...

bool	issymmetric
	Symmetry flag. internal symmetry flag. More...

double[]	d
	Arrays for internal storage of eigenvalues. internal storage of eigenvalues. More...

double[]	e

double[][]	V
	Array for internal storage of eigenvectors. internal storage of eigenvectors. More...

double[][]	H
	Array for internal storage of nonsymmetric Hessenberg form. internal storage of nonsymmetric Hessenberg form. More...

double[]	ort
	Working storage for nonsymmetric algorithm. working storage for nonsymmetric algorithm. More...

double	cdivr

double	cdivi

Detailed Description

Eigenvalues and eigenvectors of a real matrix. If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal. I.e. A = V.Multiply(D.Multiply(V.Transpose())) and V.Multiply(V.Transpose()) equals the identity matrix. If A is not symmetric, then the eigenvalue matrix D is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda]. The columns of V represent the eigenvectors in the sense that A*V = V*D, i.e. A.Multiply(V) equals V.Multiply(D). The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*Inverse(V) depends upon V.cond().

Constructor & Destructor Documentation

DotNetMatrix.EigenvalueDecomposition.EigenvalueDecomposition ( GeneralMatrix Arg )

inline

Check for symmetry, then construct the eigenvalue decomposition

Parameters

Arg	Square matrix

Returns: Structure to access D and V.

References DotNetMatrix.GeneralMatrix.Array, DotNetMatrix.GeneralMatrix.ColumnDimension, DotNetMatrix.EigenvalueDecomposition.hqr2(), DotNetMatrix.EigenvalueDecomposition.n, DotNetMatrix.EigenvalueDecomposition.orthes(), DotNetMatrix.EigenvalueDecomposition.tql2(), and DotNetMatrix.EigenvalueDecomposition.tred2().

DotNetMatrix.EigenvalueDecomposition.EigenvalueDecomposition ( GeneralMatrix Arg )

inline

Check for symmetry, then construct the eigenvalue decomposition

Parameters

Arg	Square matrix

Returns: Structure to access D and V.

References DotNetMatrix.GeneralMatrix.Array, DotNetMatrix.GeneralMatrix.ColumnDimension, DotNetMatrix.EigenvalueDecomposition.hqr2(), DotNetMatrix.EigenvalueDecomposition.n, DotNetMatrix.EigenvalueDecomposition.orthes(), DotNetMatrix.EigenvalueDecomposition.tql2(), and DotNetMatrix.EigenvalueDecomposition.tred2().

Member Function Documentation

void DotNetMatrix.EigenvalueDecomposition.tred2 ( )

inlineprivate

References DotNetMatrix.EigenvalueDecomposition.n.

Referenced by DotNetMatrix.EigenvalueDecomposition.EigenvalueDecomposition().

void DotNetMatrix.EigenvalueDecomposition.tql2 ( )

inlineprivate

References DotNetMatrix.EigenvalueDecomposition.n.

Referenced by DotNetMatrix.EigenvalueDecomposition.EigenvalueDecomposition().

void DotNetMatrix.EigenvalueDecomposition.orthes ( )

inlineprivate

References DotNetMatrix.EigenvalueDecomposition.n.

Referenced by DotNetMatrix.EigenvalueDecomposition.EigenvalueDecomposition().

void DotNetMatrix.EigenvalueDecomposition.cdiv	(	double	xr,
		double	xi,
		double	yr,
		double	yi
	)

inlineprivate

References DotNetMatrix.EigenvalueDecomposition.d.

Referenced by DotNetMatrix.EigenvalueDecomposition.hqr2().

void DotNetMatrix.EigenvalueDecomposition.hqr2 ( )

inlineprivate

References DotNetMatrix.EigenvalueDecomposition.cdiv(), DotNetMatrix.EigenvalueDecomposition.cdivi, DotNetMatrix.EigenvalueDecomposition.cdivr, and DotNetMatrix.EigenvalueDecomposition.n.

Referenced by DotNetMatrix.EigenvalueDecomposition.EigenvalueDecomposition().

virtual GeneralMatrix DotNetMatrix.EigenvalueDecomposition.GetV ( )

inlinevirtual

Return the eigenvector matrix

Returns: V

Referenced by DotNetMatrix.test.TestMatrix.Main().

void ISerializable. DotNetMatrix.EigenvalueDecomposition.GetObjectData	(	SerializationInfo	info,
		StreamingContext	context
	)

inlineprivate

void DotNetMatrix.EigenvalueDecomposition.tred2 ( )

inlineprivate

References DotNetMatrix.EigenvalueDecomposition.n.

void DotNetMatrix.EigenvalueDecomposition.tql2 ( )

inlineprivate

References DotNetMatrix.EigenvalueDecomposition.n.

void DotNetMatrix.EigenvalueDecomposition.orthes ( )

inlineprivate

References DotNetMatrix.EigenvalueDecomposition.n.

void DotNetMatrix.EigenvalueDecomposition.cdiv	(	double	xr,
		double	xi,
		double	yr,
		double	yi
	)

inlineprivate

References DotNetMatrix.EigenvalueDecomposition.d.

void DotNetMatrix.EigenvalueDecomposition.hqr2 ( )

inlineprivate

References DotNetMatrix.EigenvalueDecomposition.cdiv(), DotNetMatrix.EigenvalueDecomposition.cdivi, DotNetMatrix.EigenvalueDecomposition.cdivr, and DotNetMatrix.EigenvalueDecomposition.n.

virtual GeneralMatrix DotNetMatrix.EigenvalueDecomposition.GetV ( )

inlinevirtual

Return the eigenvector matrix

Returns: V

void ISerializable. DotNetMatrix.EigenvalueDecomposition.GetObjectData	(	SerializationInfo	info,
		StreamingContext	context
	)

inlineprivate

Member Data Documentation

int DotNetMatrix.EigenvalueDecomposition.n

private

Row and column dimension (square matrix). matrix dimension.

Referenced by DotNetMatrix.EigenvalueDecomposition.EigenvalueDecomposition(), DotNetMatrix.EigenvalueDecomposition.hqr2(), DotNetMatrix.EigenvalueDecomposition.orthes(), DotNetMatrix.EigenvalueDecomposition.tql2(), and DotNetMatrix.EigenvalueDecomposition.tred2().

bool DotNetMatrix.EigenvalueDecomposition.issymmetric

private

Symmetry flag. internal symmetry flag.

double [] DotNetMatrix.EigenvalueDecomposition.d

private

Arrays for internal storage of eigenvalues. internal storage of eigenvalues.

Referenced by DotNetMatrix.EigenvalueDecomposition.cdiv().

double [] DotNetMatrix.EigenvalueDecomposition.e

private

double [][] DotNetMatrix.EigenvalueDecomposition.V

private

Array for internal storage of eigenvectors. internal storage of eigenvectors.

double [][] DotNetMatrix.EigenvalueDecomposition.H

private

Array for internal storage of nonsymmetric Hessenberg form. internal storage of nonsymmetric Hessenberg form.

double [] DotNetMatrix.EigenvalueDecomposition.ort

private

Working storage for nonsymmetric algorithm. working storage for nonsymmetric algorithm.

double DotNetMatrix.EigenvalueDecomposition.cdivr

private

Referenced by DotNetMatrix.EigenvalueDecomposition.hqr2().

double DotNetMatrix.EigenvalueDecomposition.cdivi

private

Referenced by DotNetMatrix.EigenvalueDecomposition.hqr2().

Property Documentation

virtual double [] DotNetMatrix.EigenvalueDecomposition.RealEigenvalues

get

Return the real parts of the eigenvalues

Returns: real(diag(D))

Referenced by DotNetMatrix.examples.MagicSquareExample.Main().

virtual double [] DotNetMatrix.EigenvalueDecomposition.ImagEigenvalues

get

Return the imaginary parts of the eigenvalues

Returns: imag(diag(D))

GeneralMatrix DotNetMatrix.EigenvalueDecomposition.D

get

Return the block diagonal eigenvalue matrix

Returns: D

Referenced by DotNetMatrix.test.TestMatrix.Main().

The documentation for this class was generated from the following file:

igtest/numerics/DotMatrix/Backup/EigenvalueDecomposition.cs

Public Member Functions

Properties

Private Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

Property Documentation