EigenMatrix (i.e. eigendecomposition or spectral decomposition) decomposes a diagonalisable nonsymmetric real square matrix into its canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. More...
Public Member Functions | |
| EigenMatrix ()=delete | |
| No default construct. More... | |
| EigenMatrix (const EigenMatrix &)=delete | |
| No copy construct. More... | |
| EigenMatrix & | operator= (const EigenMatrix &)=delete |
| No copy assignment. More... | |
| EigenMatrix (const SquareMatrix< cmptType > &A) | |
| Construct from a SquareMatrix<cmptType> More... | |
| EigenMatrix (const SquareMatrix< cmptType > &A, bool symmetric) | |
| Construct from a SquareMatrix<cmptType> and symmetry flag. More... | |
| const DiagonalMatrix< cmptType > & | EValsRe () const |
| Return real eigenvalues or real part of complex eigenvalues. More... | |
| const DiagonalMatrix< cmptType > & | EValsIm () const |
| const SquareMatrix< cmptType > & | EVecs () const |
| const SquareMatrix< complex > | complexEVecs () const |
| Return right eigenvectors in unpacked form. More... | |
EigenMatrix (i.e. eigendecomposition or spectral decomposition) decomposes a diagonalisable nonsymmetric real square matrix into its canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.
The eigenvalue equation (i.e. eigenvalue problem) is:
\[ A v = \lambda v \]
where
| \( A \) | = | a diagonalisable square matrix of dimension m-by-m |
| \( v \) | = | a (non-zero) vector of dimension m (right eigenvector) |
| \( \lambda \) | = | a scalar corresponding to v (eigenvalue) |
If A is symmetric, the following relation is satisfied:
\[ A = v*D*v^T \]
where
| \( D \) | = | diagonal real eigenvalue matrix |
| \( v \) | = | orthogonal eigenvector matrix |
If A is not symmetric, D becomes a block diagonal matrix wherein the real eigenvalues are present on the diagonal within 1-by-1 blocks, and complex eigenvalues within 2-by-2 blocks, i.e. \(\lambda + i \mu\) with \([\lambda, \mu; -\mu, \lambda]\).
The columns of v represent eigenvectors corresponding to eigenvalues, satisfying the eigenvalue equation. Even though eigenvalues of a matrix are unique, eigenvectors of the matrix are not. For the same eigenvalue, the corresponding eigenvector can be real or complex with non-unique entries. In addition, the validity of the equation \(A = v*D*v^T\) depends on the condition number of v, which can be ill-conditioned, or singular for invalidated equations.
References:
OpenFOAM-compatible implementation:
Passalacqua, A., Heylmun, J., Icardi, M.,
Madadi, E., Bachant, P., & Hu, X. (2019).
OpenQBMM 5.0.1 for OpenFOAM 7, Zenodo.
DOI:10.5281/zenodo.3471804
Implementations for the functions:
'tridiagonaliseSymmMatrix', 'symmTridiagonalQL',
'Hessenberg' and 'realSchur' (based on ALGOL-procedure:tred2):
Wilkinson, J. H., & Reinsch, C. (1971).
In Bauer, F. L. & Householder A. S. (Eds.),
Handbook for Automatic Computation: Volume II: Linear Algebra.
(Vol. 186), Springer-Verlag Berlin Heidelberg.
DOI: 10.1007/978-3-642-86940-2
Explanations on how real eigenvectors
can be unpacked into complex domain:
Moler, C. (1998).
Re: Eigenvectors.
Retrieved from https://bit.ly/3ao4Wat
TNT/JAMA implementation:
Pozo, R. (1997).
Template Numerical Toolkit for linear algebra:
High performance programming with C++
and the Standard Template Library.
The International Journal of Supercomputer Applications
and High Performance Computing, 11(3), 251-263.
DOI:10.1177/109434209701100307
(No particular order) Hicklin, J., Moler, C., Webb, P.,
Boisvert, R. F., Miller, B., Pozo, R., & Remington, K. (2012).
JAMA: A Java Matrix Package 1.0.3.
Retrived from https://math.nist.gov/javanumerics/jama/OpenQBMM eigenSolver class (2019) without any changes to its internal mechanisms. Therefore, no differences between EigenMatrix and eigenSolver (2019) classes should be expected in terms of input-process-output operations.OpenQBMM eigenSolver class derives almost completely from the TNT/JAMA implementation, a public-domain library developed by NIST and MathWorks from 1998 to 2012, available at http://math.nist.gov/tnt/index.html (Retrieved June 6, 2020). Their implementation was based upon EISPACK.tridiagonaliseSymmMatrix, symmTridiagonalQL, Hessenberg and realSchur methods are based on the Algol procedures tred2 by Bowdler, Martin, Reinsch, and Wilkinson, Handbook for Auto. Comp., Vol. II-Linear Algebra, and the corresponding FORTRAN subroutine in EISPACK.Definition at line 178 of file EigenMatrix.H.
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delete |
No default construct.
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delete |
No copy construct.
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explicit |
Construct from a SquareMatrix<cmptType>
Definition at line 1007 of file EigenMatrix.C.
References A, Foam::abort(), Foam::FatalError, and FatalErrorInFunction.
| EigenMatrix | ( | const SquareMatrix< cmptType > & | A, |
| bool | symmetric | ||
| ) |
Construct from a SquareMatrix<cmptType> and symmetry flag.
Does not perform symmetric check
Definition at line 1039 of file EigenMatrix.C.
References A, Foam::abort(), Foam::FatalError, and FatalErrorInFunction.
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delete |
No copy assignment.
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inline |
Return real eigenvalues or real part of complex eigenvalues.
Definition at line 248 of file EigenMatrix.H.
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inline |
Return zero-matrix for real eigenvalues or imaginary part of complex eigenvalues
Definition at line 255 of file EigenMatrix.H.
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inline |
Return right eigenvectors matrix where each column is a right eigenvector that corresponds to an eigenvalue
Definition at line 262 of file EigenMatrix.H.
| const Foam::SquareMatrix< Foam::complex > complexEVecs | ( | ) | const |
Return right eigenvectors in unpacked form.
Definition at line 1076 of file EigenMatrix.C.
References Foam::roots::complex, Foam::mag(), Foam::transform(), and x.
1.8.17