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/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math.linear;
/**
* An interface to classes that implement an algorithm to calculate the
* eigen decomposition of a real matrix.
* <p>The eigen decomposition of matrix A is a set of two matrices:
* V and D such that A = V × D × V<sup>T</sup>.
* A, V and D are all m × m matrices.</p>
* <p>This interface is similar in spirit to the <code>EigenvalueDecomposition</code>
* class from the <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a>
* library, with the following changes:</p>
* <ul>
* <li>a {@link #getVT() getVt} method has been added,</li>
* <li>two {@link #getRealEigenvalue(int) getRealEigenvalue} and {@link #getImagEigenvalue(int)
* getImagEigenvalue} methods to pick up a single eigenvalue have been added,</li>
* <li>a {@link #getEigenvector(int) getEigenvector} method to pick up a single
* eigenvector has been added,</li>
* <li>a {@link #getDeterminant() getDeterminant} method has been added.</li>
* <li>a {@link #getSolver() getSolver} method has been added.</li>
* </ul>
* @see <a href="http://mathworld.wolfram.com/EigenDecomposition.html">MathWorld</a>
* @see <a href="http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix">Wikipedia</a>
* @version $Revision: 997726 $ $Date: 2010-09-16 14:39:51 +0200 (jeu. 16 sept. 2010) $
* @since 2.0
*/
public interface EigenDecomposition {
/**
* Returns the matrix V of the decomposition.
* <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
* <p>The columns of V are the eigenvectors of the original matrix.</p>
* <p>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).</p>
* @return the V matrix
*/
RealMatrix getV();
/**
* Returns the block diagonal matrix D of the decomposition.
* <p>D is a block diagonal matrix.</p>
* <p>Real eigenvalues are on the diagonal while complex values are on
* 2x2 blocks { {real +imaginary}, {-imaginary, real} }.</p>
* @return the D matrix
* @see #getRealEigenvalues()
* @see #getImagEigenvalues()
*/
RealMatrix getD();
/**
* Returns the transpose of the matrix V of the decomposition.
* <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p>
* <p>The columns of V are the eigenvectors of the original matrix.</p>
* <p>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).</p>
* @return the transpose of the V matrix
*/
RealMatrix getVT();
/**
* Returns a copy of the real parts of the eigenvalues of the original matrix.
* @return a copy of the real parts of the eigenvalues of the original matrix
* @see #getD()
* @see #getRealEigenvalue(int)
* @see #getImagEigenvalues()
*/
double[] getRealEigenvalues();
/**
* Returns the real part of the i<sup>th</sup> eigenvalue of the original matrix.
* @param i index of the eigenvalue (counting from 0)
* @return real part of the i<sup>th</sup> eigenvalue of the original matrix
* @see #getD()
* @see #getRealEigenvalues()
* @see #getImagEigenvalue(int)
*/
double getRealEigenvalue(int i);
/**
* Returns a copy of the imaginary parts of the eigenvalues of the original matrix.
* @return a copy of the imaginary parts of the eigenvalues of the original matrix
* @see #getD()
* @see #getImagEigenvalue(int)
* @see #getRealEigenvalues()
*/
double[] getImagEigenvalues();
/**
* Returns the imaginary part of the i<sup>th</sup> eigenvalue of the original matrix.
* @param i index of the eigenvalue (counting from 0)
* @return imaginary part of the i<sup>th</sup> eigenvalue of the original matrix
* @see #getD()
* @see #getImagEigenvalues()
* @see #getRealEigenvalue(int)
*/
double getImagEigenvalue(int i);
/**
* Returns a copy of the i<sup>th</sup> eigenvector of the original matrix.
* @param i index of the eigenvector (counting from 0)
* @return copy of the i<sup>th</sup> eigenvector of the original matrix
* @see #getD()
*/
RealVector getEigenvector(int i);
/**
* Return the determinant of the matrix
* @return determinant of the matrix
*/
double getDeterminant();
/**
* Get a solver for finding the A × X = B solution in exact linear sense.
* @return a solver
*/
DecompositionSolver getSolver();
}
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