de.lmu.ifi.dbs.elki.math.linearalgebra
Class EigenvalueDecomposition

java.lang.Object
  de.lmu.ifi.dbs.elki.math.linearalgebra.EigenvalueDecomposition

All Implemented Interfaces:

Serializable

public class EigenvalueDecomposition
extends Object
implements Serializable

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.times(D.timesTranspose(V)) and V.timesTranspose(V) 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.times(V) equals V.times(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().

See Also:

Serialized Form

Field Summary

private double	cdivi
private double	cdivr
private double[]	d Arrays for internal storage of eigenvalues.
private double[]	e Arrays for internal storage of eigenvalues.
private double[][]	H Array for internal storage of nonsymmetric Hessenberg form.
private boolean	issymmetric Symmetry flag.
private int	n Row and column dimension (square matrix).
private double[]	ort Working storage for nonsymmetric algorithm.
private static long	serialVersionUID Serial version
private double[][]	V Array for internal storage of eigenvectors.

Constructor Summary

EigenvalueDecomposition(Matrix Arg)
Check for symmetry, then construct the eigenvalue decomposition

Method Summary

private void	cdiv(double xr, double xi, double yr, double yi)
Matrix	getD() Return the block diagonal eigenvalue matrix
double[]	getImagEigenvalues() Return the imaginary parts of the eigenvalues
double[]	getRealEigenvalues() Return the real parts of the eigenvalues
Matrix	getV() Return the eigenvector matrix
private void	hqr2()
private void	orthes()
private void	tql2()
private void	tred2()

Methods inherited from class java.lang.Object

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

Field Detail

serialVersionUID

private static final long serialVersionUID

Serial version

See Also:

Constant Field Values

n

private int n

Row and column dimension (square matrix).

issymmetric

private boolean issymmetric

Symmetry flag.

d

private double[] d

Arrays for internal storage of eigenvalues.

e

private double[] e

Arrays for internal storage of eigenvalues.

V

private double[][] V

Array for internal storage of eigenvectors.

H

private double[][] H

Array for internal storage of nonsymmetric Hessenberg form.

ort

private double[] ort

Working storage for nonsymmetric algorithm.

cdivr

private transient double cdivr

cdivi

private transient double cdivi

Constructor Detail

EigenvalueDecomposition

public EigenvalueDecomposition(Matrix Arg)

Check for symmetry, then construct the eigenvalue decomposition

Parameters:

Arg - Square matrix

Method Detail

tred2

private void tred2()

tql2

private void tql2()

orthes

private void orthes()

cdiv

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

hqr2

private void hqr2()

getV

public Matrix getV()

Return the eigenvector matrix

Returns:

V

getRealEigenvalues

public double[] getRealEigenvalues()

Return the real parts of the eigenvalues

Returns:

real(diag(D))

getImagEigenvalues

public double[] getImagEigenvalues()

Return the imaginary parts of the eigenvalues

Returns:

imag(diag(D))

getD

public Matrix getD()

Return the block diagonal eigenvalue matrix

Returns:

D