# Class EigenDecompositionNonSymmetric

java.lang.Object
org.hipparchus.linear.EigenDecompositionNonSymmetric

public class EigenDecompositionNonSymmetric extends Object
Calculates the eigen decomposition of a non-symmetric real matrix.

The eigen decomposition of matrix A is a set of two matrices: $$V$$ and $$D$$ such that $$A V = V D$$ where \$$$A$$, $$V$$ and $$D$$ are all $$m \times m$$ matrices.

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

This class supports non-symmetric matrices, which have complex eigenvalues. Support for symmetric matrices is provided by EigenDecompositionSymmetric.

As $$A$$ is not symmetric, then the eigenvalue matrix $$D$$ is block diagonal with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues, $$\lambda \pm i \mu$$, in 2-by-2 blocks:

$\begin{bmatrix} \lambda & \mu\\ -\mu & \lambda \end{bmatrix}$

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 $$A = V D V^{-1}$$ 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:
3.0
• ## Field Summary

Fields
Modifier and Type
Field
Description
static final double
DEFAULT_EPSILON
Default epsilon value to use for internal epsilon
• ## Constructor Summary

Constructors
Constructor
Description
EigenDecompositionNonSymmetric(RealMatrix matrix)
Calculates the eigen decomposition of the given real matrix.
EigenDecompositionNonSymmetric(RealMatrix matrix, double epsilon)
Calculates the eigen decomposition of the given real matrix.
• ## Method Summary

Modifier and Type
Method
Description
RealMatrix
getD()
Gets the block diagonal matrix D of the decomposition.
Complex
getDeterminant()
Computes the determinant of the matrix.
Complex
getEigenvalue(int i)
Returns the ith eigenvalue of the original matrix.
Complex[]
getEigenvalues()
Gets a copy of the eigenvalues of the original matrix.
FieldVector<Complex>
getEigenvector(int i)
Gets a copy of the ith eigenvector of the original matrix.
double
getEpsilon()
Get's the value for epsilon which is used for internal tests (e.g.
RealMatrix
getV()
Gets the matrix V of the decomposition.
RealMatrix
getVInv()
Gets the inverse of the matrix V of the decomposition.

### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ## Field Details

• ### DEFAULT_EPSILON

public static final double DEFAULT_EPSILON
Default epsilon value to use for internal epsilon
• ## Constructor Details

• ### EigenDecompositionNonSymmetric

public EigenDecompositionNonSymmetric(RealMatrix matrix)
Calculates the eigen decomposition of the given real matrix.
Parameters:
matrix - Matrix to decompose.
Throws:
MathIllegalStateException - if the algorithm fails to converge.
MathRuntimeException - if the decomposition of a general matrix results in a matrix with zero norm
• ### EigenDecompositionNonSymmetric

public EigenDecompositionNonSymmetric(RealMatrix matrix, double epsilon) throws MathRuntimeException
Calculates the eigen decomposition of the given real matrix.
Parameters:
matrix - Matrix to decompose.
epsilon - Epsilon used for internal tests (e.g. is singular, eigenvalue ratio, etc.)
Throws:
MathIllegalStateException - if the algorithm fails to converge.
MathRuntimeException - if the decomposition of a general matrix results in a matrix with zero norm
• ## Method Details

• ### getV

public RealMatrix getV()
Gets the matrix V of the decomposition. V is a matrix whose columns hold either the real or the imaginary part of eigenvectors.
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.
• ### getEpsilon

public double getEpsilon()
Get's the value for epsilon which is used for internal tests (e.g. is singular, eigenvalue ratio, etc.)
Returns:
the epsilon value.
• ### getVInv

public RealMatrix getVInv()
Gets the inverse of the matrix V of the decomposition.
Returns:
the inverse of the V matrix.
• ### getEigenvalues

public Complex[] getEigenvalues()
Gets a copy of the eigenvalues of the original matrix.
Returns:
a copy of the eigenvalues of the original matrix.
• ### getEigenvalue

public Complex getEigenvalue(int i)
Returns the ith eigenvalue of the original matrix.
Parameters:
i - index of the eigenvalue (counting from 0)
Returns:
ith eigenvalue of the original matrix.
• ### getEigenvector

public  getEigenvector(int i)
Gets a copy of the ith eigenvector of the original matrix.

Note that if the the ith is complex this method will throw an exception.

Parameters:
i - Index of the eigenvector (counting from 0).
Returns:
a copy of the ith eigenvector of the original matrix.