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eigendecomposition Class Reference

Eigen decomposition of a square matrix. More...

Public Member Functions

 eigendecomposition (const scalarSquareMatrix &A)
 Construct the eigen decomposition of matrix A. More...
 
const scalarField & d () const
 Return the real part of the eigenvalues. More...
 
const scalarField & e () const
 Return the imaginary part of the eigenvalues. More...
 
void D (scalarSquareMatrix &D) const
 Return the block diagonal eigenvalue matrix in D. More...
 
const scalarSquareMatrix & V () const
 Return the eigenvector matrix. More...
 

Detailed Description

Eigen decomposition of a square matrix.

Calculates the eigenvalues and eigenvectors of a square scalar matrix matrix.

If the matrix $ A $ is symmetric, then $ A = VDV^-1 $ 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 $ VV^-1 = I $, where $ I $ is the identity matrix. The columns of V represent the eigenvectors in the sense that $ AV = VD $.

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 + ib $, 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 $ AV = VD $.

The matrix $ V $ may be badly conditioned, or even singular, so the validity of the equation $ A = VDV^-1 $ depends upon the condition number of $ V $.

Adapted from the TNT C++ implementation (http://math.nist.gov/tnt) of the JAMA Java implementation (http://math.nist.gov/javanumerics/jama) of the algorithms in: 

    Wilkinson, J. H., & Reinsch, C. (1971).
    Handbook for Automatic Computation: Volume II:
    Linear Algebra (Vol. 186). Springer-Verlag.
Source files

Definition at line 95 of file eigendecomposition.H.

Constructor & Destructor Documentation

◆ eigendecomposition()

eigendecomposition ( const scalarSquareMatrix &  A)

Construct the eigen decomposition of matrix A.

Definition at line 1050 of file eigendecomposition.C.

References Foam::saturationModels::A, n, and List< T >::setSize().

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Member Function Documentation

◆ d()

const scalarField& d ( ) const
inline

Return the real part of the eigenvalues.

Definition at line 157 of file eigendecomposition.H.

◆ e()

const scalarField& e ( ) const
inline

Return the imaginary part of the eigenvalues.

Definition at line 163 of file eigendecomposition.H.

◆ D()

void D ( scalarSquareMatrix &  D) const

Return the block diagonal eigenvalue matrix in D.

Definition at line 1107 of file eigendecomposition.C.

References n, and SquareMatrix< Type >::setSize().

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◆ V()

const scalarSquareMatrix& V ( ) const
inline

Return the eigenvector matrix.

Definition at line 172 of file eigendecomposition.H.

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