TemplateJAMA::Eigenvalue class Reference
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JAMA::Eigenvalue Class Template Reference
#include
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
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
const TNT::Array2D< Real > & A ) [inline]
Check for symmetry, then construct the eigenvalue decomposition
Parameters:
Square real (non-complex) matrix
Member Function Documentation
template
void JAMA::Eigenvalue
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:
upon return, the matrix is filled with the block diagonal eigenvalue matrix.
template
void JAMA::Eigenvalue
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
TNT::Array1D< Real > & d_ ) [inline]
Return the real parts of the eigenvalues
real(diag(D))
template
void JAMA::Eigenvalue
TNT::Array2D< Real > & V_ ) [inline]
Return the eigenvector matrix
The documentation for this class was generated from the following file: jama_eig.h
Generated at Mon Jan 20 07:47:18 2003 for JAMA/C++ by
1.2.5 written by Dimitri van Heesch,
© 1997-2001
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