TNT::Linear_Algebra::Eigenvalue< Real > Class Template Reference

#include <tnt_linalg.h>

Public Member Functions
	Eigenvalue (const Matrix< Real > &A)
void	getV (Matrix< Real > &V_)
void	getRealEigenvalues (Vector< Real > &d_)
void	getImagEigenvalues (Vector< Real > &e_)
void	getD (Matrix< Real > &D)

Detailed Description

template<class Real>
class TNT::Linear_Algebra::Eigenvalue< Real >

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), which in turn, were based on original EISPACK routines.

Constructor & Destructor Documentation

Eigenvalue()

template<class Real >

TNT::Linear_Algebra::Eigenvalue< Real >::Eigenvalue ( const Matrix< Real > & A )

inline

Check for symmetry, then construct the eigenvalue decomposition

Parameters

A	Square real (non-complex) matrix

Member Function Documentation

getD()

template<class Real >

void TNT::Linear_Algebra::Eigenvalue< Real >::getD ( Matrix< Real > & 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.

getImagEigenvalues()

template<class Real >

void TNT::Linear_Algebra::Eigenvalue< Real >::getImagEigenvalues ( Vector< Real > & e_ )

inline

Return the imaginary parts of the eigenvalues in parameter e_.

Parameters

e_	new matrix with imaginary parts of the eigenvalues.

getRealEigenvalues()

template<class Real >

void TNT::Linear_Algebra::Eigenvalue< Real >::getRealEigenvalues ( Vector< Real > & d_ )

inline

Return the real parts of the eigenvalues

Returns

real(diag(D))

getV()

template<class Real >

void TNT::Linear_Algebra::Eigenvalue< Real >::getV ( Matrix< Real > & V_ )

inline

Return the eigenvector matrix

Returns

V

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The documentation for this class was generated from the following file:

• tnt_linalg.h