Eigenvalue< ValueType > Class Template Reference

#include <Eigenvalue.h>

Public Types
typedef boost::numeric::ublas::matrix < ValueType >	Matrix
typedef boost::numeric::ublas::vector < ValueType >	Vector

Public Member Functions
	Eigenvalue (const Matrix &A)
const Matrix &	getV () const
const Vector &	getRealEigenvalues () const
const Vector &	getImagEigenvalues () const
void	getD (Matrix &D)

Static Public Member Functions
static bool	isSymmetric (const Matrix &A)

Detailed Description

template<class ValueType>
class Eigenvalue< ValueType >

Computes eigenvalues and eigenvectors of a real (non-complex) matrix.

If A is symmetric, then A = V*D*V' where the eigenvalue matrix D is diagonal and the eigenvector matrix V is orthogonal. That is, the diagonal values of D are the eigenvalues, and V*V' = I, where I is the identity matrix. The columns of V represent the eigenvectors in the sense that A*V = V*D.

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, a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex eigenvalues look like

          u + iv     .        .          .      .    .
            .      u - iv     .          .      .    .
            .        .      a + ib       .      .    .
            .        .        .        a - ib   .    .
            .        .        .          .      x    .
            .        .        .          .      .    y

then D looks like

            u        v        .          .      .    .
           -v        u        .          .      .    . 
            .        .        a          b      .    .
            .        .       -b          a      .    .
            .        .        .          .      x    .
            .        .        .          .      .    y

This keeps V a real matrix in both symmetric and non-symmetric cases, and A*V = V*D.

The matrix V may be badly conditioned, or even singular, so the validity of the equation A = V*D*inverse(V) depends upon the condition number of V.

(Adapted from JAMA, a Java Matrix Library, developed by jointly 
by the Mathworks and NIST; see  http://math.nist.gov/javanumerics/jama).

Member Typedef Documentation

template<class ValueType>

typedef boost::numeric::ublas::matrix<ValueType> Eigenvalue< ValueType >::Matrix

template<class ValueType>

typedef boost::numeric::ublas::vector<ValueType> Eigenvalue< ValueType >::Vector

Constructor & Destructor Documentation

template<class ValueType>

Eigenvalue< ValueType >::Eigenvalue ( const Matrix &  A )

inline

Check for symmetry, then construct the eigenvalue decomposition

Parameters

A	Square real (non-complex) matrix

References Eigenvalue< ValueType >::isSymmetric().

Member Function Documentation

template<class ValueType>

void Eigenvalue< ValueType >::getD ( Matrix &  D )

inline

Computes the block diagonal eigenvalue matrix. If the original matrix 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, a + i*b, in 2-by-2 blocks, [a, b; -b, a]. That is, if the complex eigenvalues look like

          u + iv     .        .          .      .    .
            .      u - iv     .          .      .    .
            .        .      a + ib       .      .    .
            .        .        .        a - ib   .    .
            .        .        .          .      x    .
            .        .        .          .      .    y

then D looks like

            u        v        .          .      .    .
           -v        u        .          .      .    . 
            .        .        a          b      .    .
            .        .       -b          a      .    .
            .        .        .          .      x    .
            .        .        .          .      .    y

This keeps V a real matrix in both symmetric and non-symmetric cases, and A*V = V*D.

Parameters

D	upon return, the matrix is filled with the block diagonal eigenvalue matrix.

template<class ValueType>

const Vector& Eigenvalue< ValueType >::getImagEigenvalues ( ) const

inline

Return the imaginary parts of the eigenvalues

Returns

: imaginary parts of the eigenvalues

template<class ValueType>

const Vector& Eigenvalue< ValueType >::getRealEigenvalues ( ) const

inline

Return the real parts of the eigenvalues

Returns

real(diag(D))

Referenced by CMDS< ValueType >::CMDS().

template<class ValueType>

const Matrix& Eigenvalue< ValueType >::getV ( ) const

inline

Return the eigenvector matrix

Returns

V

Referenced by CMDS< ValueType >::CMDS().

template<class ValueType>

static bool Eigenvalue< ValueType >::isSymmetric ( const Matrix &  A )

inlinestatic

This function is used to decide wether the procedure for symmetric matrices can be used.

Referenced by Eigenvalue< ValueType >::Eigenvalue().