1 /* 2 * Licensed to the Apache Software Foundation (ASF) under one or more 3 * contributor license agreements. See the NOTICE file distributed with 4 * this work for additional information regarding copyright ownership. 5 * The ASF licenses this file to You under the Apache License, Version 2.0 6 * (the "License"); you may not use this file except in compliance with 7 * the License. You may obtain a copy of the License at 8 * 9 * http://www.apache.org/licenses/LICENSE-2.0 10 * 11 * Unless required by applicable law or agreed to in writing, software 12 * distributed under the License is distributed on an "AS IS" BASIS, 13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 14 * See the License for the specific language governing permissions and 15 * limitations under the License. 16 */ 17 18 package org.apache.commons.math.linear; 19 20 21 /** 22 * An interface to classes that implement an algorithm to calculate the 23 * eigen decomposition of a real matrix. 24 * <p>The eigen decomposition of matrix A is a set of two matrices: 25 * V and D such that A = V × D × V<sup>T</sup>. 26 * A, V and D are all m × m matrices.</p> 27 * <p>This interface is similar in spirit to the <code>EigenvalueDecomposition</code> 28 * class from the <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> 29 * library, with the following changes:</p> 30 * <ul> 31 * <li>a {@link #getVT() getVt} method has been added,</li> 32 * <li>two {@link #getRealEigenvalue(int) getRealEigenvalue} and {@link #getImagEigenvalue(int) 33 * getImagEigenvalue} methods to pick up a single eigenvalue have been added,</li> 34 * <li>a {@link #getEigenvector(int) getEigenvector} method to pick up a single 35 * eigenvector has been added,</li> 36 * <li>a {@link #getDeterminant() getDeterminant} method has been added.</li> 37 * <li>a {@link #getSolver() getSolver} method has been added.</li> 38 * </ul> 39 * @see <a href="http://mathworld.wolfram.com/EigenDecomposition.html">MathWorld</a> 40 * @see <a href="http://en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix">Wikipedia</a> 41 * @version $Revision: 997726 $ $Date: 2010-09-16 14:39:51 +0200 (jeu. 16 sept. 2010) $ 42 * @since 2.0 43 */ 44 public interface EigenDecomposition { 45 46 /** 47 * Returns the matrix V of the decomposition. 48 * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p> 49 * <p>The columns of V are the eigenvectors of the original matrix.</p> 50 * <p>No assumption is made about the orientation of the system axes formed 51 * by the columns of V (e.g. in a 3-dimension space, V can form a left- 52 * or right-handed system).</p> 53 * @return the V matrix 54 */ 55 RealMatrix getV(); 56 57 /** 58 * Returns the block diagonal matrix D of the decomposition. 59 * <p>D is a block diagonal matrix.</p> 60 * <p>Real eigenvalues are on the diagonal while complex values are on 61 * 2x2 blocks { {real +imaginary}, {-imaginary, real} }.</p> 62 * @return the D matrix 63 * @see #getRealEigenvalues() 64 * @see #getImagEigenvalues() 65 */ 66 RealMatrix getD(); 67 68 /** 69 * Returns the transpose of the matrix V of the decomposition. 70 * <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p> 71 * <p>The columns of V are the eigenvectors of the original matrix.</p> 72 * <p>No assumption is made about the orientation of the system axes formed 73 * by the columns of V (e.g. in a 3-dimension space, V can form a left- 74 * or right-handed system).</p> 75 * @return the transpose of the V matrix 76 */ 77 RealMatrix getVT(); 78 79 /** 80 * Returns a copy of the real parts of the eigenvalues of the original matrix. 81 * @return a copy of the real parts of the eigenvalues of the original matrix 82 * @see #getD() 83 * @see #getRealEigenvalue(int) 84 * @see #getImagEigenvalues() 85 */ 86 double[] getRealEigenvalues(); 87 88 /** 89 * Returns the real part of the i<sup>th</sup> eigenvalue of the original matrix. 90 * @param i index of the eigenvalue (counting from 0) 91 * @return real part of the i<sup>th</sup> eigenvalue of the original matrix 92 * @see #getD() 93 * @see #getRealEigenvalues() 94 * @see #getImagEigenvalue(int) 95 */ 96 double getRealEigenvalue(int i); 97 98 /** 99 * Returns a copy of the imaginary parts of the eigenvalues of the original matrix. 100 * @return a copy of the imaginary parts of the eigenvalues of the original matrix 101 * @see #getD() 102 * @see #getImagEigenvalue(int) 103 * @see #getRealEigenvalues() 104 */ 105 double[] getImagEigenvalues(); 106 107 /** 108 * Returns the imaginary part of the i<sup>th</sup> eigenvalue of the original matrix. 109 * @param i index of the eigenvalue (counting from 0) 110 * @return imaginary part of the i<sup>th</sup> eigenvalue of the original matrix 111 * @see #getD() 112 * @see #getImagEigenvalues() 113 * @see #getRealEigenvalue(int) 114 */ 115 double getImagEigenvalue(int i); 116 117 /** 118 * Returns a copy of the i<sup>th</sup> eigenvector of the original matrix. 119 * @param i index of the eigenvector (counting from 0) 120 * @return copy of the i<sup>th</sup> eigenvector of the original matrix 121 * @see #getD() 122 */ 123 RealVector getEigenvector(int i); 124 125 /** 126 * Return the determinant of the matrix 127 * @return determinant of the matrix 128 */ 129 double getDeterminant(); 130 131 /** 132 * Get a solver for finding the A × X = B solution in exact linear sense. 133 * @return a solver 134 */ 135 DecompositionSolver getSolver(); 136 137 } 138
