org.apache.commons.math.linear

Interface EigenDecomposition

All Known Implementing Classes:

EigenDecompositionImpl

public interface EigenDecomposition

An interface to classes that implement an algorithm to calculate the eigen decomposition of a real matrix.

The eigen decomposition of matrix A is a set of two matrices: V and D such that A = V × D × V^T. A, V and D are all m × m matrices.

This interface is similar in spirit to the EigenvalueDecomposition class from the JAMA library, with the following changes:

• a getVt method has been added,
• two getRealEigenvalue and getImagEigenvalue methods to pick up a single eigenvalue have been added,
• a getEigenvector method to pick up a single eigenvector has been added,
• a getDeterminant method has been added.
• a getSolver method has been added.

Since:

2.0

See Also:

MathWorld, Wikipedia

Method Summary

Modifier and Type	Method and Description
RealMatrix	getD() Returns the block diagonal matrix D of the decomposition.
double	getDeterminant() Return the determinant of the matrix
RealVector	getEigenvector(int i) Returns a copy of the ith eigenvector of the original matrix.
double	getImagEigenvalue(int i) Returns the imaginary part of the ith eigenvalue of the original matrix.
double[]	getImagEigenvalues() Returns a copy of the imaginary parts of the eigenvalues of the original matrix.
double	getRealEigenvalue(int i) Returns the real part of the ith eigenvalue of the original matrix.
double[]	getRealEigenvalues() Returns a copy of the real parts of the eigenvalues of the original matrix.
DecompositionSolver	getSolver() Get a solver for finding the A × X = B solution in exact linear sense.
RealMatrix	getV() Returns the matrix V of the decomposition.
RealMatrix	getVT() Returns the transpose of the matrix V of the decomposition.

Method Detail

getV

RealMatrix getV()

Returns the matrix V of the decomposition.

V is an orthogonal matrix, i.e. its transpose is also its inverse.

The columns of V are the eigenvectors of the original matrix.

No assumption is made about the orientation of the system axes formed by the columns of V (e.g. in a 3-dimension space, V can form a left- or right-handed system).

Returns:

the V matrix

getD

RealMatrix getD()

Returns the block diagonal matrix D of the decomposition.

D is a block diagonal matrix.

Real eigenvalues are on the diagonal while complex values are on 2x2 blocks { {real +imaginary}, {-imaginary, real} }.

Returns:

the D matrix

See Also:

getRealEigenvalues(), getImagEigenvalues()

getVT

RealMatrix getVT()

Returns the transpose of the matrix V of the decomposition.

V is an orthogonal matrix, i.e. its transpose is also its inverse.

The columns of V are the eigenvectors of the original matrix.

No assumption is made about the orientation of the system axes formed by the columns of V (e.g. in a 3-dimension space, V can form a left- or right-handed system).

Returns:

the transpose of the V matrix

getRealEigenvalues

double[] getRealEigenvalues()

Returns a copy of the real parts of the eigenvalues of the original matrix.

Returns:

a copy of the real parts of the eigenvalues of the original matrix

See Also:

getD(), getRealEigenvalue(int), getImagEigenvalues()

getRealEigenvalue

double getRealEigenvalue(int i)

Returns the real part of the i^th eigenvalue of the original matrix.

Parameters:

i - index of the eigenvalue (counting from 0)

Returns:

real part of the i^th eigenvalue of the original matrix

See Also:

getD(), getRealEigenvalues(), getImagEigenvalue(int)

getImagEigenvalues

double[] getImagEigenvalues()

Returns a copy of the imaginary parts of the eigenvalues of the original matrix.

Returns:

a copy of the imaginary parts of the eigenvalues of the original matrix

See Also:

getD(), getImagEigenvalue(int), getRealEigenvalues()

getImagEigenvalue

double getImagEigenvalue(int i)

Returns the imaginary part of the i^th eigenvalue of the original matrix.

Parameters:

i - index of the eigenvalue (counting from 0)

Returns:

imaginary part of the i^th eigenvalue of the original matrix

See Also:

getD(), getImagEigenvalues(), getRealEigenvalue(int)

getEigenvector

RealVector getEigenvector(int i)

Returns a copy of the i^th eigenvector of the original matrix.

Parameters:

i - index of the eigenvector (counting from 0)

Returns:

copy of the i^th eigenvector of the original matrix

See Also:

getD()

getDeterminant

double getDeterminant()

Return the determinant of the matrix

Returns:

determinant of the matrix

getSolver

DecompositionSolver getSolver()

Get a solver for finding the A × X = B solution in exact linear sense.

Returns:

a solver