org.opensourcephysics.numerics

Class EigenvalueDecomposition

• java.lang.Object
• org.opensourcephysics.numerics.EigenvalueDecomposition
• All Implemented Interfaces:
java.io.Serializable

```public class EigenvalueDecomposition
extends java.lang.Object
implements java.io.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.times(V.transpose())) and V.times(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.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
• Constructor Summary

Constructors
Constructor and Description
`EigenvalueDecomposition(double[][] A)`
Constructs the EigenvalueDecomposition.
• Method Summary

Methods
Modifier and Type Method and Description
`double[][]` `getEigenvalueMatrix()`
Return the block diagonal eigenvalue matrix
`double[][]` `getEigenvectors()`
Return the eigenvector matrix
`double[]` `getImagEigenvalues()`
Return the imaginary parts of the eigenvalues
`double[]` `getRealEigenvalues()`
Return the real parts of the eigenvalues
`static double` ```hypot(double a, double b)```
sqrt(a^2 + b^2) without under/overflow.
`void` `recompute(double[][] A)`
Check for symmetry, then recompute the eigenvalue decomposition
• Methods inherited from class java.lang.Object

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

• EigenvalueDecomposition

`public EigenvalueDecomposition(double[][] A)`
Constructs the EigenvalueDecomposition. Allocates arrays, checks for symmetry and then constructs the eigenvalue decomposition
Parameters:
`A` - square matrix
• Method Detail

• recompute

`public void recompute(double[][] A)`
Check for symmetry, then recompute the eigenvalue decomposition
Parameters:
`A` - square matrix
• getEigenvectors

`public double[][] getEigenvectors()`
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))
• getEigenvalueMatrix

`public double[][] getEigenvalueMatrix()`
Return the block diagonal eigenvalue matrix
Returns:
D
• hypot

```public static double hypot(double a,
double b)```
sqrt(a^2 + b^2) without under/overflow.