<commit_before>/**
 * Copyright (C) 2010 openecho
 *
 * Licensed 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.
 * 
 * This class contains portions of code which are Copyright The Mathworks and
 * NIST. Their license statement follows,
 *
 * --------------------------------------------------------------------------
 *
 * This software is a cooperative product of The MathWorks and the National
 * Institute of Standards and Technology (NIST) which has been released to the
 * public domain. Neither The MathWorks nor NIST assumes any responsibility
 * whatsoever for its use by other parties, and makes no guarantees, expressed
 * or implied, about its quality, reliability, or any other characteristic.
 *
 * Copyright (C) 1999 The Mathworks and NIST
 **/
package kinetic.math.linear;

import kinetic.math.Matrix;
import kinetic.math.MatrixD;
import kinetic.math.QuickMath;

/** 
 * Decomposes a real matrix A into its eigenvalues and eigenvectors.
 * <P>
 * 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.
 * <P>
 * 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().
 *
 * @author openecho
 * @author The Mathworks and NIST
 **/<commit_msg>Comment update<commit_after>/**
 * Copyright (C) 2010 openecho
 *
 * Licensed 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.
 * 
 * This class contains portions of code which are Copyright The Mathworks and
 * NIST. Their license statement follows,
 *
 * --------------------------------------------------------------------------
 *
 * This software is a cooperative product of The MathWorks and the National
 * Institute of Standards and Technology (NIST) which has been released to the
 * public domain. Neither The MathWorks nor NIST assumes any responsibility
 * whatsoever for its use by other parties, and makes no guarantees, expressed
 * or implied, about its quality, reliability, or any other characteristic.
 *
 * Copyright (C) 1999 The Mathworks and NIST
 **/
package kinetic.math.linear;

import kinetic.math.Matrix;
import kinetic.math.MatrixD;
import kinetic.math.QuickMath;

/** 
 * Decomposes a real matrix A into its eigenvalues and eigenvectors
 *
 * 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().
 *
 * @author openecho
 * @author The Mathworks and NIST
 **/
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