## Class EigenDecomposition

• public class EigenDecomposition
extends Object
Calculates 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 × VT. A, V and D are all m × m matrices.

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

As of 3.1, this class supports general real matrices (both symmetric and non-symmetric):

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 the condition of V.

This implementation is based on the paper by A. Drubrulle, R.S. Martin and J.H. Wilkinson "The Implicit QL Algorithm" in Wilksinson and Reinsch (1971) Handbook for automatic computation, vol. 2, Linear algebra, Springer-Verlag, New-York.

Since:
2.0 (changed to concrete class in 3.0)
MathWorld, Wikipedia
• ### Constructor Summary

Constructors
Constructor Description
EigenDecomposition​(double[] main, double[] secondary)
Calculates the eigen decomposition of the symmetric tridiagonal matrix.
EigenDecomposition​(RealMatrix matrix)
Calculates the eigen decomposition of the given real matrix.
• ### Method Summary

All Methods
Modifier and Type Method Description
RealMatrix getD()
Gets the block diagonal matrix D of the decomposition.
double getDeterminant()
Computes the determinant of the matrix.
RealVector getEigenvector​(int i)
Gets a copy of the ith eigenvector of the original matrix.
double getImagEigenvalue​(int i)
Gets the imaginary part of the ith eigenvalue of the original matrix.
double[] getImagEigenvalues()
Gets 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()
Gets a copy of the real parts of the eigenvalues of the original matrix.
DecompositionSolver getSolver()
Gets a solver for finding the A × X = B solution in exact linear sense.
RealMatrix getSquareRoot()
Computes the square-root of the matrix.
RealMatrix getV()
Gets the matrix V of the decomposition.
RealMatrix getVT()
Gets the transpose of the matrix V of the decomposition.
boolean hasComplexEigenvalues()
Returns whether the calculated eigen values are complex or real.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### EigenDecomposition

public EigenDecomposition​(RealMatrix matrix)
throws MathArithmeticException
Calculates the eigen decomposition of the given real matrix.

Supports decomposition of a general matrix since 3.1.

Parameters:
matrix - Matrix to decompose.
Throws:
MaxCountExceededException - if the algorithm fails to converge.
MathArithmeticException - if the decomposition of a general matrix results in a matrix with zero norm
Since:
3.1
• #### EigenDecomposition

public EigenDecomposition​(double[] main,
double[] secondary)
Calculates the eigen decomposition of the symmetric tridiagonal matrix. The Householder matrix is assumed to be the identity matrix.
Parameters:
main - Main diagonal of the symmetric tridiagonal form.
secondary - Secondary of the tridiagonal form.
Throws:
MaxCountExceededException - if the algorithm fails to converge.
Since:
3.1
• ### Method Detail

• #### getV

public RealMatrix getV()
Gets 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

public RealMatrix getD()
Gets 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.
getRealEigenvalues(), getImagEigenvalues()
• #### getVT

public RealMatrix getVT()
Gets 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.
• #### hasComplexEigenvalues

public boolean hasComplexEigenvalues()
Returns whether the calculated eigen values are complex or real.

The method performs a zero check for each element of the getImagEigenvalues() array and returns true if any element is not equal to zero.

Returns:
true if the eigen values are complex, false otherwise
Since:
3.1
• #### getEigenvector

public RealVector getEigenvector​(int i)
Gets a copy of the ith eigenvector of the original matrix.
Parameters:
i - Index of the eigenvector (counting from 0).
Returns:
a copy of the ith eigenvector of the original matrix.
getD()
• #### getDeterminant

public double getDeterminant()
Computes the determinant of the matrix.
Returns:
the determinant of the matrix.
• #### getSquareRoot

public RealMatrix getSquareRoot()
Computes the square-root of the matrix. This implementation assumes that the matrix is symmetric and positive definite.
Returns:
the square-root of the matrix.
Throws:
MathUnsupportedOperationException - if the matrix is not symmetric or not positive definite.
Since:
3.1
• #### getSolver

public DecompositionSolver getSolver()
Gets a solver for finding the A × X = B solution in exact linear sense.

Since 3.1, eigen decomposition of a general matrix is supported, but the DecompositionSolver only supports real eigenvalues.

Returns:
a solver
Throws:
MathUnsupportedOperationException - if the decomposition resulted in complex eigenvalues