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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)

## 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).

## Constructor & Destructor Documentation

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

Check for symmetry, then construct the eigenvalue decomposition

Parameters:
 A Square real (non-complex) matrix

## Member Function Documentation

 template void JAMA::Eigenvalue::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 void JAMA::Eigenvalue::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 void JAMA::Eigenvalue::getRealEigenvalues ( TNT::Array1D< Real > & d_ )` [inline]`
 Return the real parts of the eigenvalues Returns: real(diag(D))

 template void JAMA::Eigenvalue::getV ( TNT::Array2D< Real > & V_ )` [inline]`
 Return the eigenvector matrix Returns: V

The documentation for this class was generated from the following file:
Generated at Mon Jan 20 07:47:18 2003 for JAMA/C++ by 1.2.5 written by Dimitri van Heesch, © 1997-2001