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JAMA::Eigenvalue Class Template Reference

#include <jama_eig.h>

List of all members.

Public Methods
	Eigenvalue (const TNT::Array2D< Real > &A)
void	getV (TNT::Array2D< Real > &V_)
void	getRealEigenvalues (TNT::Array1D< Real > &d_)
void	getImagEigenvalues (TNT::Array1D< Real > &e_)
void	getD (TNT::Array2D< Real > &D)

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Detailed Description

template<class Real> class JAMA::Eigenvalue

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).

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Constructor & Destructor Documentation

template<class Real> JAMA::Eigenvalue<Real>::Eigenvalue<Real> ( const TNT::Array2D< Real > & A ) [inline]

Check for symmetry, then construct the eigenvalue decomposition Parameters: A   Square real (non-complex) matrix

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Member Function Documentation

template<class Real> void JAMA::Eigenvalue<Real>::getD ( TNT::Array2D< 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.

template<class Real> void JAMA::Eigenvalue<Real>::getImagEigenvalues ( TNT::Array1D< Real > & e_ ) [inline]

Return the imaginary parts of the eigenvalues in parameter e_. arm e_: new matrix with imaginary parts of the eigenvalues.

template<class Real> void JAMA::Eigenvalue<Real>::getRealEigenvalues ( TNT::Array1D< Real > & d_ ) [inline]

Return the real parts of the eigenvalues Returns: real(diag(D))

template<class Real> void JAMA::Eigenvalue<Real>::getV ( TNT::Array2D< Real > & V_ ) [inline]

Return the eigenvector matrix Returns: V

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

• jama_eig.h

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