org.jeometry.math.decomposition.EigenDecomposition Maven / Gradle / Ivy

Go to download

Show more of this group Show more artifacts with this name
Show all versions of jeometry-api Show documentation

Jeometry, a Mathematic and Geometry library for Java

There is a newer version: 1.0.5

package org.jeometry.math.decomposition;

import org.jeometry.Jeometry;
import org.jeometry.math.Matrix;

/**
 *  This interface describes an Eigen decomposition.

 *  Let A be a square matrix. According to the Eigen decomposition theorem, we can write:

 *  A = VDV^-1

 *  Where:
 *  
 *  V is a square matrix made of eigenvectors
 *  
D is a diagonal matrix made of eigenvalues
 *  
 *  If V is not a square matrix, then it cannot have a matrix inverse and A does not have an eigen decomposition. 
 *  In this case, if V is m×n (with m>n), then A can be written using a {@link SVDDecomposition Singular Values Decomposition}. 
 * 
 *  If A is symmetric, then: 

 *  
A = VDV^T


 *  where D is diagonal and V is orthogonal (VV^T = I).
 * 
 *  If A is not symmetric, then the eigenvalue matrix D is block diagonal
 *  with:
 *  

 *  the real eigenvalues in 1-by-1 blocks
 *  
any complex eigenvalues λ + i×μ in 2-by-2 blocks [λ, μ; -μ, λ]
 *  
 *  
 *  The columns of V represent the eigenvectors in the sense that:

 *  AV = VD
 *  

 *  The matrix V may be badly conditioned, or even singular, so the validity of the equation cannot be sure for all matrices.
 * @author Julien Seinturier - COMEX S.A. - [email protected] - https://github.com/jorigin/jeometry
 * @version {@value Jeometry#version} b{@value Jeometry#BUILD}
 * @since 1.0.0
 */
public interface EigenDecomposition extends Decomposition {
	
	/**
	 * The index of the eigenvectors V matrix within the {@link Decomposition#getComponents() decomposition components}. 
	 */
	public static final int COMPONENT_V_INDEX = 0;
	
	/**
	 * The index of the diagonal eigenvalues D matrix within the {@link Decomposition#getComponents() decomposition components}. 
	 */
	public static final int COMPONENT_D_INDEX = 1;
	
	/**
	 * Get the diagonal matrix of eigenvalues, denoted D.
	 * @return  the diagonal matrix of eigenvalues, denoted D
	 */
	public Matrix getD();
	
	/**
	 * Get the matrix of eigenvectors, denoted V.
	 * @return the matrix of eigenvectors, denoted V 
	 */
	public Matrix getV();
}