java.lang.Object
- org.apache.commons.math.linear.EigenDecompositionImpl

All Implemented Interfaces:

EigenDecomposition
```
public class EigenDecompositionImpl
extends Object
implements EigenDecomposition
```
Calculates the eigen decomposition of a real symmetric 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.

As of 2.0, this class supports only symmetric matrices, and hence computes only real realEigenvalues. This implies the D matrix returned by getD() is always diagonal and the imaginary values returned getImagEigenvalue(int) and getImagEigenvalues() are always null.

When called with a RealMatrix argument, this implementation only uses the upper part of the matrix, the part below the diagonal is not accessed at all.

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

Constructor Summary

Constructors
Constructor	Description
`EigenDecompositionImpl(double[] main, double[] secondary, double splitTolerance)`	Calculates the eigen decomposition of the symmetric tridiagonal matrix.
`EigenDecompositionImpl(RealMatrix matrix, double splitTolerance)`	Calculates the eigen decomposition of the given symmetric matrix.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method	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 i^th eigenvector of the original matrix.
`double`	`getImagEigenvalue(int i)`	Returns the imaginary part of the i^th 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 i^th 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.

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - EigenDecompositionImpl
```
public EigenDecompositionImpl(RealMatrix matrix,
                              double splitTolerance)
                       throws InvalidMatrixException
```
    Calculates the eigen decomposition of the given symmetric matrix.
    
    Parameters:
    
    matrix - The symmetric matrix to decompose.
    
    splitTolerance - dummy parameter, present for backward compatibility only.
    
    Throws:
    
    InvalidMatrixException - (wrapping a ConvergenceException if algorithm fails to converge
  - EigenDecompositionImpl
```
public EigenDecompositionImpl(double[] main,
                              double[] secondary,
                              double splitTolerance)
                       throws InvalidMatrixException
```
    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 triadiagonal form
    
    secondary - Secondary of the tridiagonal form
    
    splitTolerance - dummy parameter, present for backward compatibility only.
    
    Throws:
    
    InvalidMatrixException - (wrapping a ConvergenceException if algorithm fails to converge
- Method Detail
  - getV
```
public RealMatrix getV()
                throws InvalidMatrixException
```
    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).
    
    Specified by:
    
    getV in interface EigenDecomposition
    
    Returns:
    
    the V matrix
    
    Throws:
    
    InvalidMatrixException
  - getD
```
public RealMatrix getD()
                throws InvalidMatrixException
```
    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} }.
    
    Specified by:
    
    getD in interface EigenDecomposition
    
    Returns:
    
    the D matrix
    
    Throws:
    
    InvalidMatrixException
    
    See Also:
    
    EigenDecomposition.getRealEigenvalues(), EigenDecomposition.getImagEigenvalues()
  - getVT
```
public RealMatrix getVT()
                 throws InvalidMatrixException
```
    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).
    
    Specified by:
    
    getVT in interface EigenDecomposition
    
    Returns:
    
    the transpose of the V matrix
    
    Throws:
    
    InvalidMatrixException
  - getRealEigenvalues
```
public double[] getRealEigenvalues()
                            throws InvalidMatrixException
```
    Returns a copy of the real parts of the eigenvalues of the original matrix.
    
    Specified by:
    
    getRealEigenvalues in interface EigenDecomposition
    
    Returns:
    
    a copy of the real parts of the eigenvalues of the original matrix
    
    Throws:
    
    InvalidMatrixException
    
    See Also:
    
    EigenDecomposition.getD(), EigenDecomposition.getRealEigenvalue(int), EigenDecomposition.getImagEigenvalues()
  - getRealEigenvalue
```
public double getRealEigenvalue(int i)
                         throws InvalidMatrixException,
                                ArrayIndexOutOfBoundsException
```
    Returns the real part of the i^th eigenvalue of the original matrix.
    
    Specified by:
    
    getRealEigenvalue in interface EigenDecomposition
    
    Parameters:
    
    i - index of the eigenvalue (counting from 0)
    
    Returns:
    
    real part of the i^th eigenvalue of the original matrix
    
    Throws:
    
    InvalidMatrixException
    
    ArrayIndexOutOfBoundsException
    
    See Also:
    
    EigenDecomposition.getD(), EigenDecomposition.getRealEigenvalues(), EigenDecomposition.getImagEigenvalue(int)
  - getImagEigenvalues
```
public double[] getImagEigenvalues()
                            throws InvalidMatrixException
```
    Returns a copy of the imaginary parts of the eigenvalues of the original matrix.
    
    Specified by:
    
    getImagEigenvalues in interface EigenDecomposition
    
    Returns:
    
    a copy of the imaginary parts of the eigenvalues of the original matrix
    
    Throws:
    
    InvalidMatrixException
    
    See Also:
    
    EigenDecomposition.getD(), EigenDecomposition.getImagEigenvalue(int), EigenDecomposition.getRealEigenvalues()
  - getImagEigenvalue
```
public double getImagEigenvalue(int i)
                         throws InvalidMatrixException,
                                ArrayIndexOutOfBoundsException
```
    Returns the imaginary part of the i^th eigenvalue of the original matrix.
    
    Specified by:
    
    getImagEigenvalue in interface EigenDecomposition
    
    Parameters:
    
    i - index of the eigenvalue (counting from 0)
    
    Returns:
    
    imaginary part of the i^th eigenvalue of the original matrix
    
    Throws:
    
    InvalidMatrixException
    
    ArrayIndexOutOfBoundsException
    
    See Also:
    
    EigenDecomposition.getD(), EigenDecomposition.getImagEigenvalues(), EigenDecomposition.getRealEigenvalue(int)
  - getEigenvector
```
public RealVector getEigenvector(int i)
                          throws InvalidMatrixException,
                                 ArrayIndexOutOfBoundsException
```
    Returns a copy of the i^th eigenvector of the original matrix.
    
    Specified by:
    
    getEigenvector in interface EigenDecomposition
    
    Parameters:
    
    i - index of the eigenvector (counting from 0)
    
    Returns:
    
    copy of the i^th eigenvector of the original matrix
    
    Throws:
    
    InvalidMatrixException
    
    ArrayIndexOutOfBoundsException
    
    See Also:
    
    EigenDecomposition.getD()
  - getDeterminant
```
public double getDeterminant()
```
    Return the determinant of the matrix
    
    Specified by:
    
    getDeterminant in interface EigenDecomposition
    
    Returns:
    
    determinant of the matrix
  - getSolver
```
public DecompositionSolver getSolver()
```
    Get a solver for finding the A × X = B solution in exact linear sense.
    
    Specified by:
    
    getSolver in interface EigenDecomposition
    
    Returns:
    
    a solver

Class EigenDecompositionImpl

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

EigenDecompositionImpl

EigenDecompositionImpl

Method Detail

getV

getD

getVT

getRealEigenvalues

getRealEigenvalue

getImagEigenvalues

getImagEigenvalue

getEigenvector

getDeterminant

getSolver